Hilbert Between the Formal and the Informal Side of Mathematics

Abstract

In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution.

Hilbert's Synthesis on Foundation of Geometry

“…Some different hypotheses have been advanced on the reason for its adjunction [e.g. 12,74]. We can here suppose Hilbert adds it to reconcile his axiomatic treatment with the one of Riemann, Helmholtz and Lie to account for the motions of rigid bodies, defined by them as continuous spatial transformations.…”

Imagination And Creative Thinking Skills Of Elementary School Students In Learning Mathematics: A Reflection Of Realistic Mathematics Education

“…This is because the thinking stage of elementary school students is still not formal, even elementary school students in lower grades are still in the pre-concrete thinking stage (Kholiq, 2020). On the other hand, mathematics is a deductive, axiomatic, formal, hierarchical, abstract science, and is a meaningful language of symbols (Venturi, 2015;Vojkuvkova, 2012). Given these differences in characteristics, it is necessary to have a special ability from a teacher to bridge the world of children who have not thought deductively in order to understand the deductive world of mathematics.…”

“…Imagination is the power of thought to imagine (in wishful thinking) or create images (paintings, essays, etc.) of events based on reality or one's general experience (Pelaprat & Cole, 2011;Venturi, 2015). With imagination, humans develop something from simplicity to be more valuable in mind (Pelaprat & Cole, 2011).…”

Análise das condições de verdade e dos requerimentos existenciais em axiomatizações da aritmética

“…71 73 Esta proposta resolve diversas questões delicadas em filosofia da matemática de modo extremamente rude. 74 Com relação a verdade das sentenças aritméticas, a predicação de 1 + 3 = 2 + 2 deve ser fruto da transformação dos símbolos primitivos pela aplicação das regras do jogo-aritmética. Uma vez que, matematicamente, a obtenção de uma sentença pela aplicação das regras do jogo aos símbolos é a demonstração da sentença, a veracidade de 1 + 3 = 2 + 2 é identificada com sua demonstrabilidade, e o mesmo cânone deve ser aplicado tanto ao último teorema de Fermat quanto à qualquer outra sentença aritmética.…”

“…[33, p. 5]. 74 O problema metafísico da existência de números é resolvido pela negação de sua importância e relevância; não há números e, mesmo que existam, não são relevantes à aritmética. A segunda parte dessa resposta carece de plausibilidade, pois negar a relevância dos números para a aritmética contradiz a história, a prática e a aplicabilidade desta disciplina.…”