Deciding the Undecidable: Wrestling with Hilbert's Problems (original) (raw)

Solved and unsolved mathematical problems

Lecture Notes from the year 2014 in the subject Mathematics - General, Basics, language: English, abstract: In der folgenden Ausarbeitung werden einige zentrale mathematische Probleme und ihre Losungsansatze beschrieben, u.a.: Zahlensysteme, Ungeloste Probleme der Antike, Befreundete Zahlen und pythagoreische Triple, Vollkommene Zahlen, Der Goldene Schnitt, Wahrscheinlichkeitsrechnung nach Pascal, Fibonacci-Zahlen, Goldbach-Vermutung, Primzahlen, Mobius-Funktion.

Mathematical Impossibility in History

Theorems stating that something is impossible are notoriously difficult to understand for many students and amateur mathematicians. In this talk I shall discuss how the role of such impossibility statements has changed during the history of mathematics. I shall argue that impossibility statements have changed status from a kind of meta-statement to true mathematical theorems. I shall also argue that this story is worth telling in the classroom because it will clarify the nature of impossibility theorems and thus of mathematics. In particular it will show to the students how mathematics is able to investigate the limits of its own activity with its own methods.

Some reflections on the problems and their role in the development of mathematics

Revista Pesquisa Qualitativa, 2020

In this work we show the historical significance of mathematical problems and their meaning, not only for Mathematics itself, but for the Philosophy of Mathematics and other related sciences, in which the current conceptual and methodological apparatus has been developed, as a result of the development of investigations related to the resolution of certain problems. From these results, we build the Mathematical Development Matrix. Keywords: Problems; History of Mathematics; Mathematics.

Is Mathematics Problem Solving or Theorem Proving?

Foundations of Science, vol. 22 (2017), pp. 183-199., 2017

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable.

The Problem of Mathematical Truth

In current mathematical practice, mathematical knowledge (if it is achieved at all) is achieved by proving theorems on the basis of definitions and axioms. The problem is to understand how what is achieved thereby constitutes knowledge; more specifically, it is to develop a unified account of mathematical truth and mathematical knowledge, one that reveals their inner connection. What stands in our way, according to a very familiar argument of Benacerraf's, is that in mathematics there seems no way to combine a Tarskian semantics, according to which truth involves ineliminable reference to objects (either by way of singular terms or by way of quantifiers), with an adequate epistemology: either mathematical knowledge is by way of proof, in which case mathematical objects are irrelevant to mathematical knowledge and then we have no account of mathematical truth, or mathematical knowledge is not by way of proof because mathematical objects are constitutive of mathematical truth, but then we have no resources for understanding mathematical knowledge. I then trace the difficulties, in a series of stages, all the way down to our most basic conception of logic as formal and merely explicative: if mathematics is a practice of reasoning from concepts by logic alone then it ought, according to Kant, to be analytic, that is, merely explicative, not knowledge properly speaking at all. This, I submit, is the really hard problem of mathematical truth. Four responses are outlined, but only one holds out promise of resolving our difficulties, namely, that of Peirce and Frege. According to them, logic is a science, and hence experimental and fallible, symbolic language is contentful despite involving no reference to any objects, and proof is a constructive and hence fruitful process. Adequately developed, these ideas will enable us finally to resolve the problem of mathematical truth.