Constructive Mathematics Is Seemingly Simple but There Are Still Open Problems: Kreisel's Observation Explained

In his correspondence with Grigory Mints, the famous logician Georg Kreisel noticed that many results of constructive mathematics seem easierto-prove than the corresponding classical (non-constructive) results – although he noted that these results are still far from being simple and the corresponding open problems are challenging. In this paper, we provide a possible explanation for this empirical observation. 1 Constructive Mathematics and Kreisel’s Observation: A Brief Introduction to the Problem Main objectives of science and engineering. In this paper, we deal with Kreisel’s observation about the so-called constructive mathematics. In order to understand what is constructive mathematics, we need to recall what are the main objectives of science and engineering. One of the main objectives of science is to predict the future state of the world: we want to predict tomorrow’s weather, we want to predict what will happen to a building during an earthquake of a certain strength, etc. Once we learn to predict what will happen in the future, the next task is how to change the future so that it will be more beneficial to us. Generally speaking, this is already the task not for science but for engineering. One of the main objective of engineering is: • to design objects with that will make it possible to achieve the desired result, and • to come up with control strategies that will lead to this result.