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I have studied that Gauss was one of the firsts mathematicians to defend this idea, about the Abstract Math and the conception of number, claiming that "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."

This thought may have been influenced (albeit indirectly) by a fellow-country philosopher of Gauss, Immanuel Kant, once in an article from a time not far from Gauss, Kant argued that the idea behind negative numbers should go beyond the traditional logical idea.

In "Attempt to introduce the conception of Negative Quantities into Philosophy", Kant puts in balance the logical-contradictory and real-oppositional ideas that the negative numbers have. As, for example, the logical idea of negative numbers attests to the fact that if a body is in motion, the negation of this movement would be rest. However, there would be no conditions for this body to be at rest and in motion simultaneously. On the other hand, the real-oppositional idea holds that the denial of a movement of a body would be the same movement in the opposite direction, and, finally, the simultaneous events of these two events would bring rest.

That is, the focus in the understanding of number is not about the things you are observing, but about the relationships between each other.

What mathematicians and philosophers between the 18th and 19th centuries were defenders or precursors of this kind of abstract mathematics and number as a relation?

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  • $\begingroup$ It might be worth keeping in mind that a good part of Gauss' job security was due to his expert development of surveying techniques, including mean-square treatment of measurement errors and such. That's the kind of thing that endears one to the local honchos... making a safe facade behind which to follow less practical matters! $\endgroup$ – paul garrett Feb 28 '18 at 0:42
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    $\begingroup$ "fellow-country philosopher of Gauss, Immanuel Kant"? Brunswick-Wolfenbüttel and Prussia were allies, but they weren't the same country - at least, not in the lifetime of either Kant or Gauss. $\endgroup$ – Peter Taylor Feb 28 '18 at 14:47
  • $\begingroup$ There were too many things packed into one question, I edited it to make it more answerable here. $\endgroup$ – Conifold Mar 3 '18 at 0:21
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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." 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. He 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 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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