Rough automatic draft, low quality: expect mistakes and awkward wording. It will be replaced by a better translation. Original →
25
It is interesting that the American mathematician M. Goldberg managed to discover in 1967 that Malfatti circles for any triangle do not provide a solution to the “original Malfatti problem” (Problem II).
Used in the consideration of the Malfatti problem, let us assume that “recourse to the limiting case” is often and successfully used to test mathematical propositions for verisimilitude.
Is it just for verification? Is it only in mathematics?
For about six years now, it seems to have turned out that “testing for verisimilitude” can only lead to the verification of dubious hypotheses. But is it not possible to arrive at the result of a “positive” characteristic in this way? Was it possible to obtain a new discovery using these methods? What about other sciences: physics, chemistry, biology? After all, any experiment, strictly speaking, can either refute a false hypothesis or confirm its veracity, but in no way prove it! And yet, we know many “laws of science” obtained experimentally (as they say, “inductively”).
All this, of course, requires a special discussion. In order not to make such a conversation dependent on the continuation of this article, we recommend to the reader the excellent books of György Pólya:
- Mathematics and truthful reasoning (IL, 1957);
- How to solve a problem (Uchrediz, 1959);
- Mathematical discovery (“Science”, 1970).
The reader can also find individual examples related to the topic of this article in our book “Mathematics after Lessons” (“Enlightenment”, 1971).
Number triangle
| | | | | | | 65 | 50 | 66 | 37 | 51 | 67 |
| :— | :— | :— | :— | :— | :— | :— | :— | :— | :— | :— | :— |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
Here you do not see the entire number triangle, but only its “left” part. The law of triangle designation is very visible, and to come in this way to the result of a “positive” characteristic? Was it possible to obtain a new discovery using these methods? What about other sciences: physics, chemistry, biology? After all, any experiment, strictly speaking, can either refute a false hypothesis or confirm its veracity, but in no way prove it! And yet, we know many “laws of science” obtained experimentally (as they say, “inductively”).
All this, of course, requires a special discussion. In order not to make such a conversation dependent on the continuation of this article, we recommend to the reader the excellent books of György Pólya:
- Mathematics and truthful reasoning (IL, 1957);
- How to solve a problem (Uchrediz, 1959);
- Mathematical discovery (“Science”, 1970).
The reader can also find individual examples related to the topic of this article in our book “Mathematics after Lessons” (“Enlightenment”, 1971).
V. Berezin