Ações de Formação de Professores de Matemática e o Movimento de Construção de sua Identidade Profissional

Considerando a importância de buscar espaços diferenciados de formação inicial de professores de matemática, para além das tradicionais disciplinas, no presente artigo foram analisadas duas ações de formação, desenvolvidas na Universidade Estadual de Londrina – UEL, nomeadamente: a exploração de tarefas para o trabalho com multiplicação e divisão de inteiros, a partir de ideias presentes nos trabalhos de René Descartes e David Hilbert; e a construção e a utilização de casos multimídia na perspectiva do Ensino Exploratório. As negociações de significados, como mecanismo da aprendizagem docente, promovidas por essas ações em um espaço colaborativo permitiram intervenção contínua dos futuros professores em um processo de dar e receber, de influenciar e ser influenciado e, por conseguinte, o movimento de constituição de sua identidade profissional.

“I have Learned Non-Euclidean Geometry Just from this Book” - Some facets of the correspondence between Friedrich Engel and David Hilbert

Friedrich Engel and David Hilbert learned to know each other at Leipzig in 1885 and exchanged letters in particular during the next 15 years which contain interesting information on the academic life of mathematicians at the end of the 19th century. In the present article we will mainly discuss a statement by Hilbert himself on Moritz Pasch’s influence on his views of geometry, and on personnel politics concerning Hermann Minkowski and Eduard Study but also Engel himself.

Join the community, with Hilbert’s twenty-three problems

“Join the community, with Hilbert’s twenty-three problems” tells the story of German mathematician David Hilbert who, in 1900 at the International Congress of Mathematicians in Paris, presented twenty-three problems in a first serious effort by any mathematician to curate a list of important open problems across mathematical subfields. In a speech accompanying his offering, he challenged mathematicians to solve all twenty-three problems in the next century. This chapter offers highlights and explanations of some of the twenty-three problems, including the Continuum Hypothesis, the Riemann Hypothesis, and Kepler’s Sphere-Packing Conjecture. With his broad reach, Hilbert understood the value of articulating community-wide mathematical goals, as well as the unifying effect that keeping score might have. Mathematics students and enthusiasts learn that they are part of a community of “zealous and enthusiastic disciples” of mathematics. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Intensionality: From Philosophical Logic to Metamathematics

The article is devoted to the study of the status of intensionality in the exact contexts of logical and mathematical theories. The emergence of intensionality in logical and mathematical discourse leads to significant obstacles in its formalization due to the appearance of indirect contexts, the uncertainty of its indication in the theoretical apparatus, as well as the presence of various kinds of difficult-to-account semantic distinctions. The refusal to consider intensionality in logic is connected with Bertrand Russell’s criticism of Alexius Meinong’s intensionality ontology, and with Willard Van Orman Quine’s criticism of the concept of meaning and quantification of modalities. It is shown that this criticism is based on a preference for the theory of indication over the theory of meaning, in terms of the distinction “Bedeutung” and “Sinn” introduced by Gottlob Frege. The extensionality thesis is explicated; by analogy with it the intensionality thesis is constructed. It is shown that complete parallelism is not possible here, and therefore we should proceed from finding cases of extensionality violation. Since the construction of formal logical systems is to a certain extent connected with the programs of the foundations of mathematics, the complex interweaving of philosophical and purely technical questions makes the question of the role of intensionality in mathematics quite confusing. However, there is one clue here: programs in the foundations of mathematics have given rise to metamathematics, which, although it stands alone, is considered a branch of mathematics. It is not by chance that, judging by the problems arising in connection with intensionality, there is a growing suspicion that intensionality can play a significant role in metamathematics. As for the question of the sense in which metamathematics results can be considered mathematical, in terms of the presence of intensional contexts in both disciplines, it is a matter of taste: for example, the autonomy of mathematical knowledge as a result of the desire of mathematicians to eliminate the influence of philosophy that took place in the case of David Hilbert may be worth considering in the context of mathematics. Thus, the rather vague concept of intensionality receives various explications in different contexts, whether it is philosophical logic or metamathematics. In any case, the detection of context intensionality is always associated with a clear narrowing of the research area. It is obvious that the creation of a more general theory of intensionality is possible within a more general framework, in which logic and mathematics must be combined. In this respect, we can hope for the resumption of a logical project, which would be a purely logical consideration made of the natural and the mathematical.

Cosmology and Hilbert's sixth problem

There have been tantalizing indications from many quarters of physical cosmology that we are living in the multiverse - a huge set of cosmological domains ("universes"). What is the structure of this larger whole is an entirely open problem on the interface between physics and metaphysics. A goal of the present paper is to draw attention to the connection between this problem and an old and celebrated puzzle in mathematical physics. Among the unresolved problems David Hilbert posed in 1900 as a challenge for the dawning century, none is more philosophically controversial than the Sixth Problem, requiring the axiomatization of physical theories. In the new century and the new millennium, this problem has remained a challenge, usually swept under the rug as "not belonging to mathematics" (as if that impacts its epistemical status) or simply "unresolved". Recent radical ontological/cosmological hypothesis of Max Tegmark, identifying mathematical and physical structures, might shed some new light onto this allegedly antiquated subject: it might be the case that the problem has already been solved, insofar we have formalized mathematical structures! While this can be seen as "cutting the Gordian knot" rather than patiently resolving the issue, we suggest that there are several advantages to taking Tegmark's solution seriously, notably in the domain of (future) physics of the observer.

El enfoque epistemológico de David Hilbert: el a priori del conocimiento y el papel de la lógica en la fundamentación de la ciencia

This paper explores the main philosophical approaches of David Hilbert’s theory of proof. Specifically, it is focuses on his ideas regarding logic, the concept of proof, the axiomatic, the concept of truth, metamathematics, the a priori knowledge and the general nature of scientific knowledge. The aim is to show and characterize his epistemological approach on the foundation of knowledge, where logic appears as a guarantee of that foundation. Hilbert supposes that the propositional apriorism, proposed by him to support mathematics, sustains — on its turn — a general method for the treatment of the problem in other areas such as natural sciences. This method is axiomatic. Broadly speaking, we intend to recover and update the Hilbert’s philosophical thinking about the role of logic for scientific knowledge.

Hilbertova aritmetizace geometrie

Tato práce se podrobně věnuje způsobu, jakým David Hilbert (1862–1943) pojal aritmetizaci geometrie v knize Grundlagen der Geometrie z roku 1899. Nejprve stručně představíme Hilbertovy předchůdce z téže doby, kteří buď po změnách v založení geometrie volali, nebo je již sami prostřednictvím axiomaticko-deduktivní metody zapracovali. Neopomeneme přitom, co dílu předcházelo v dřívějších Hilbertových přednáškách. Následně se pokusíme nastínit­ obsah prvních dvou kapitol knihy a vysvětlit dobové i věcné souvislosti, nutné k jejich pochopení. Představíme způsob implicitních definic základních pojmů a vztahů v axiomech, a dále Hilbertovo rozdělení axiomů do skupin, přičemž se zejména zaměříme na axiomy spojitosti v kontextu s otázkou o její bezespornosti. K tomu popíšeme konstrukci aritmetického modelu axiomů geometrie, který Hilbert pro důkaz bezespornosti používá. V závěru se pokusíme nastínit hlavní důvody, které Hilberta k napsání díla vedly, a některé klíčové důsledky jeho pojetí axiomatiky geometrie.

"Una mancha en el sol de Euclides": Hilbert y la teoría euclídea de las proporciones

El artículo examina una de las contribuciones más importantes a los fundamentos de la geometría euclídea elemental lograda por David Hilbert en su obra Fundamentos de la geometría (1899), a saber: la reconstrucción de la teoría euclídea de las proporciones y de los triángulos semejantes. Se argumenta que dicha reconstrucción no sólo estuvo motivada por la identificación de Hilbert de suposiciones implícitas en la teoría de Euclides, sino que además estuvo esencialmente ligada a la preocupación por la 'pureza del método'. Más aún, se afirma que, en este caso específico, el requerimiento de Hilbert por la pureza del método posee un carácter general o fundacional, esto es, no se refiere a la demostración de un teorema en particular, sino más bien es planteado respecto de la construcción axiomática de la teoría misma.