# Were Gauss's 1846 remarks on the distinction between right and left related to orientability of surfaces?

Recently i was striked by a quotation of Gauss from a letter to his student Gerling from the date June 23, 1846. This letter states in very concise words that the distinction between right-handed and left-handed 3D cartesian coordinates system is not of aprioric nature; it cannot be determined by the first principles of spatial logic. According to what i read up to now, Gauss remarked this in order to state his disagreement with Kant's philosophy of space and of space perception (Gauss expressed similar disagreement with Kant in his private letters on non-euclidean geometry). Gauss's relevant remarks are those:

One cannot reduce to concepts the distinction between two systems of three straight lines each (directed lines, of which the one system points forward, upward to the right, the other forward, upward to the left) but one can only demonstrate by holding to actually concrete spatial things. Two minds cannot reach agreement about it unless their views connect up with one and the same system present in the real world.

and there is also this quote from his letter to Schumacher (Februar 8 ,1846):

The distinction between right and left cannot be defined, but only shown, so that it is thereby a case similar to sweet and bitter...two such minds, however, cannot make themselves directly understood concerning right and left unless one and the same individual thing forms a bridge between them... i find in it the striking refutation of Kant's imagination that space is merely the form of our external perception.

The reason why i was striked by reading Gauss's quotes is that it anticipates- not in the mathematical details, but rather in the philosophical level - the notion of orientability of surfaces, and perhaps also - though not in the mathematical details - the possibility of non-orientable surfaces and spaces like Mobius's strip and it's higher dimensional analogs. As an illustration of the impossibility of drawing a distinction between right and left at certain spaces, i added here this gif:

A crab walking on a Mobius strip cannot decide between right and left.

What i'd like to know is:

• Can anyone tell something about the state of "philosophy of right and left" at Gauss's time? because perhaps i'm misinterpreting Gauss's letter and he didn't really stood against Kant's view.
• Can anyone tell a little bit about the background to this correspondence of Gauss?
• Are you sure this was in disagreement? That space and time are irreducible to concepts was sort of Kant's point with pure intuition and synthetic a priori. He even states something similarly sounding about the angle sum theorem, that it can not be extracted from the concept of triangle, only "shown" by constructing it in pure intuition. His well-known disagreement with Kant was over pure vs empirical intuition, a priori vs a posteriori, but it is kind of tangential in these fragments. – Conifold Feb 15 at 23:19
• For clearer statements of his disagreement see Was Kant a factor in forming Gauss's abstract view of mathematical objects? What he says about the non-conceptual nature of right/left cuts against what he wrote earlier in Disquisitiones Arithmeticae (1799) about the relational nature of mathematics, and in the direction towards Kant. In 1816 review he endorsed Kant on this point, writing that definitions and syllogisms "put forth only sterile blossoms unless the fertilizing living intuition of the object itself prevails everywhere". – Conifold Feb 15 at 23:30
• First of all thanks for the usefull link! On p.316-317 of Dunnington's biography of Gauss (i took this quotations from these pages), Dunnington writes in reference to these letters that Gauss disagreed with Kant. Dunnington continues and says they differed in the manner of founding a theory of space, and especially disagreed on the point that space is merely the form of our external perception. By the way, this letter appears on volume 8 of Gauss work in the section: foundations of geometry- congruence and symmetry. – user2554 Feb 15 at 23:41
• Maybe i'm reading too much into these letters... but i just wand to find out if there was a hidden idea in Gauss's words or that he just reffered to right and left without really thinking on orientability. – user2554 Feb 15 at 23:45
• What made me suspect that Gauss thought of right and left in more mathematical context was that he was really ocuppied with the foundations of topology in the last years of his life. – user2554 Feb 15 at 23:48

The short answer to the title question is likely yes. There are two separate issues discussed in the OP, one is more philosophical on Gauss's disagreements with Kant's apriorism about space, and the other, concerning his insights into the orientation of surfaces and non-orientable surfaces. The first question is addressed to some extent in Was Kant a factor in forming Gauss's abstract view of mathematical objects?, so I will focus on the second.

Gauss corresponded with Gerling for a long time, at least since 1819, when he mentioned the angle sum defect ("Gauss-Bonnet theorem") for hyperbolic triangles in a letter to him. At the time, the main subject was non-Euclidean geometry, and this is discussed at length in What did Gauss read in the Appendix? by Abardia, Reventós and Rodríguez (the reference is to Bolyai's Appendix from 1832). Issues related to what Gauss called theorema egregium (invariance of curvature under isometries) are discussed in 1825 letters to Schumacher. Gauss's celebrated work on geometry of curved surfaces Disquisitiones Generales circa Superficies Curvas came out in 1827. His relevant writings on the subject, both published and unpublished, are discussed at length in 150 years after Gauss’ «Disquisitiones generales circa superficies curvas» by Dombrowski. Dombrowski quotes an unpublished 1825 draft of Disquisitiones Generales, which shows that Gauss possessed the concept of surface orientation in 1825 (and possibly as early as 1794). In his modernized "free translation":

"The sum of the angles of a (small) geodesic triangle Δ in a curved surface in $$\mathbb{E}^3$$ is equal to the sum of π and the oriented surface area of the spherical image of Δ, where the oriented area is taken to be positive or negative according to whether the bounda­ry of the spherical image of Δ winds around the image in the same direction or in the opposite direction as the boundary of Δ winds around Δ." (57)

Gauss then generalizes it to geodesic polygons. Dombrowski discusses the context and speculates that Gauss was led to the general statement for curved surfaces by the hyperbolic case combined with his practical experience with geodesic measurements in 1812-1822:

"The result (57) was already widely known in the special case of a deve­lopable or a spherical surface. Gauss possessed the result analogous to (57) for hyperbolic geometry (and therefore essentially for surfaces of constant negative curvature, the author) already in 1794 (see G.W. 8, p. 266), and announced it to Gerling in a letter in 1819 (see G.W. 8, p. 182). It is there­fore easy to imagine that Gauss was led to a geometrically intuitive "insight" concerning the validity of (57) for the general case of a surface of non-constant Gaussian curvature from the knowledge of this fundamental case and on the basis of his rich differential-geometric experience in geodesy (with geodesics and trigonometry spheroids, but also with questions of mapping and bending) obtained in the years 1812 to 1822.

[...] Although Gauss had studied intensively the concept of "oriented surface area" as used by him in (57) (see G.W. 8, p. 398, line 2 from below), he did not think that his corresponding studies had "matured" sufficiently... The proof (with its so impressive geometric arguments, which we have just retraced from the fragment (55)) was never published by Gauss! The reason for this lies on the one hand certainly in his self-criticism of his own above mentioned sketch of a proof for (57), and particularly of his concept of the "oriented surface area of the spherical image on a curved surface" (involved in that proof) which he has not defined with that analytic rigour usually applied by him."

This result leads to theorema egregium discussed in 1825 letters to Schumacher. The Gauss-Bonnet theorem itself is brought up in the 10 October 1846 letter to Gerling (Dombrowski's translation):

"The theorem which Mr. Schweikart mentioned to vou,that in any eeometrv the sum of all outer angles of a polygon differs from 360 by a quantity,..., which is proportional to the surface area, is the first theorem lying almost on the threshold of that theory, a theorem whose necessity I already recognized in 1794".

So the appearance of orientation related issues in the letters to Gerling and Schumacher in 1846 (connected to the older themes of his agreements and disagreements with Kant) is not surprising.

As for the non-orientable surfaces, the issue is much more murky. The "Möbius band" is described in private papers of Listing and Möbius in 1858, almost simultaneously and in similar terms, Listing's description is earlier by several months. Gauss died in 1855, but both were Gauss's students, and many historians attribute many of Listing's topological ideas to Gauss. Listing himself stated something to this effect, although without specifics. Biggs in his chapter in Möbius and his band explicitly speculates that Gauss might have been the common source here, but admits that the issue is undecidable:

"Both of them describe the construction in very similar terms. Is this another one of those istances which sometimes happen in scientific discovery, where an idea whose time is ripe appears iI1dependentiy in different places but at the same time? That is certainly a possibility. Or was there a common reason for the fact that both Möbius and Listing described the Möbius band around the same time? If the latter, then the likelihood is that the common reason was connected with the work or Gauss who, as we know, had been very interested in this kind or topic. Gauss had died in 1855, so the idea cannot have been communicated by him directly, but the possibility of some link with his work remains. I doubt that the question will ever be completely resolved."

• Thanks sincerely Conifold! obviously i voted your answer. I didn't know that Gauss wrote down a concept similar to surface orientability in his unpublished 1825 draft of Disquisitions generales. Your answer really added information to my knowledge. – user2554 Feb 17 at 7:52
• @user2554 Neither did I, until now, the 1825 draft was news to me too :) You dig deep and have a much better sense for Gauss's work, so researching your questions is always fun. – Conifold Feb 19 at 4:34