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Gauss argued for philosophical issues important to the development of mathematics, such as the identity of complex numbers, among others.

I wonder what philosophical currents influenced his thinking and what were the consequences of his defense of abstract mathematics.

Was Kant's philosophy about the magnitude of negative numbers important for this thought?

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  • I recall that Gauss was reluctant to publish his work on non Euclidean geometry because he feared the "howl of the Boeotians", by which he meant the Kantians who held to the dogma that space is Euclidean.
    – Tim kinsella
    Mar 2 '18 at 18:51
  • @Timkinsella. This involves a misunderstanding of Kant's position. See my comments to Conifold below.
    – user3017
    Mar 3 '18 at 15:31
  • Apparently Gauss read and re-read Kant; according to one count - five times.
    – Mozibur Ullah
    Mar 4 '18 at 6:22
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Gauss in Disquisitiones Arithmeticae (1799) does indeed express something close to what is now called mathematical formalism and structuralism. He writes:

"What is calculated (in the sense of things already counted) are not substances (thinkable objects for themselves), but relations between two objects counted two by two... The mathematician abstracts totally from the nature of the objects and the content of their relations; he is concerned solely with the counting and the comparison of the relations among themselves".

But it is hard to ascribe similar opinions to Kant, in fact his view was just the opposite, that concepts without intuitions are empty, and that (pure) intuitions for mathematical concepts are supplied by imaginative construction according to a priori schemata of space and time. In other words, relations are constructed only along with their intuitive objects. This led Kant to claim the a priori certainty for Euclidean geometry, something Gauss explicitly criticized him for:

"Precisely the impossibility of deciding a priori between [Euclidean and non-Euclidean space] gives the clearest proof that Kant was not justified in asserting that space is just the form of our perception." [quoted from Gauss, Kästner, and Kant by Baker].

Indeed, relationalization and formalization of mathematics in 19-th century went hand in hand with rejecting (notably by Frege and Hilbert) Kant's intuitive conception of it (which was retained more by Poincare and intuitionists).

Gauss had plenty of earlier sources to build on though. The abstractization of mathematics can be traced back to Vieta's Isagoge (1591). Bos writes in Redefining Geometrical Exactness, Ch.8:

"It was Viète who first introduced and promoted the idea that algebra was proper method for analysis of problems both in geometry, and in the theory of numbers... Viète usually reserved the term specious logistics for that part of his new algebra that dealt with abstract magnitude and in which therefore no assumptions coud be made about the actual effectuation of algebraic operations... Viète did not see algebra as a technique concerning numbers... but as a method of symbolic calculation concerning abstract magnitudes."

For more see Esteve's The Role of Symbolic Language in the transformation of Mathematics. The next thinker to advance relational/abstract view of mathematics, and Kant's chosen foil, was Leibniz. Leibniz is the one who called imaginary numbers and infinitesimals "useful fictions" and promoted the so-called "generality of algebra" (the term was coined by Cauchy), treating algebraic idenities as purely formal rules that apply regardless of the nature of the quantities involved. Peckhaus describes in Calculus Ratiocinator vs. Characteristica Universalis:

"‘Abbreviating’ means that as soon as a characteristic sign has been established for a complex object, memory can be relieved of the burden of retaining all the characteristic elements of this object. Natural languages are not sufficient for this job of designating objects unambiguously. Only in the language of arithmetic and algebra has this idea been partially realized. All reasoning in these branches consists in using characters. Errors in reasoning prove to be miscalculations... He uses arbitrarily chosen letters according to the model of mathematics. This notation allows ‘calculating with concepts’ according to sets of rules, each of them forming a calculus ratiocinator."

Frege specifically names Leibniz's calculus ratiocinator as inspiration for his concept-script (predicate calculus). Generality of algebra was later extensively used by Euler, Lagrange, and Gauss himself, in "algebraic analysis" that preceded modern Weierstrassian one, see The Foundational Aspects of Gauss's Work by Ferraro:

"Finally, eighteenth-century functions were characterised in an essential way by the use of a formal methodology that it made it possible to operate upon analytical expressions, independently of their meaning. This formal methodology was based upon two closely connected analogical principles, the generality of algebra and the extension of rules and procedures from the finite to the infinite. The generality of algebra consisted of the following assumption: (GA) if an analytical formula was derived by using the rules of algebra, then it was thought to be valid in general"."

Peacock later renamed this into Principle of Permanence of Form in his Symbolical Algebra (1831):"Whatever form is Algebraically equivalent to another, when expressed in general symbols, must be true, whatever those symbols denote." The abstractization of algebra was further promoted, before Hilbert, by Hankel and Dedekind.

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    Kant asserted, "a mathematician cannot define a place, a direction, a straight line, etc., for these are all given concepts," but all other non-primitive concepts may be arbitrarily created, so I see no reason why such flexibility could not be applied to non-euclidean geometry (afterall, spherical geometry already existed in Kant's day). Did Gauss believe you could abstract from the primitive notions as well? Can you do geometry without presupposing the notion of position, for example? Also, it's a contradiction to assert that an empirical fact could determine an a priori form of intuition.
    – user3017
    Mar 3 '18 at 15:29
  • @PédeLeão In the early 19th century, I don't think you could talk about "lines" on a sphere. I think people had a more platonic conception of these things. I.e. the geometry of the sphere did not correspond to the "true" geometry of "space". To reason about hyperbolic space in a sense one does have to redefine "line", "direction" etc. Which sounds like something Kant would want to prohibit. But idk, I'm no expert in this history-- I find it very hard to put myself in the mindset of people grappling with these issues before foundational issues were sorted out
    – Tim kinsella
    Mar 3 '18 at 20:48
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    @Timkinsella. The mindset and language usage of the people is irrelevant; spherical geometry is still non-euclidean. We intuitively understand Non-euclidean geometry within a euclidean framework. Regardless of whether we call a curve a line or not, our intuition of it is still curved, so the notion of straightness not only stands (as Kant held) but is presupposed. And it appears that you're not distinguishing between space as a form of intuition and what you call the "true geometry of space." Kant maintained: "Space is nothing other than merely the form of all appearances of outer sense."
    – user3017
    Mar 4 '18 at 1:28
  • @PédeLeão Spherical geometry in the relevant sense did not exist in Kant's day, it is a retroactive projection from modern formalistic conception of geometry. What Gauss (and later Riemann) appears to have in mind is something intermediate, "intuitive" geometry is too vague to be determined as Euclidean, it is constrained only loosely a priori and empirical input is needed to specify it fully. How much constrained is unclear in Gauss's case, Riemann though it could be just locally Euclidean, Helmholtz that it must be of constant curvature.
    – Conifold
    Mar 5 '18 at 20:39
  • @Conifold. But as I said, it's a contradiction to assert that an empirical fact could constrain an a priori form of intuition, so the only thing that makes sense is that space, as a form of intuition, must be homogeneously isometric with three dimensions. Therefore, any attribution of curvature or discreteness to space can be nothing more than an explanatory model, intuitively understood within an isometric framework, which, in fact, is what mathematicians actually do. For this reason, criticisms against Kant's notion of space are easily defeated on a purely logical basis.
    – user3017
    Mar 5 '18 at 21:38

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