IsGodAMathematician

From APIDesign

The philosphical parts of TheAPIBook were heavily influenced by The Key Stone of European Knowledge by Petr Vopěnka. I really mean inspired. The whole "key stone" book has more than 800 pages and thus TheAPIBook (of about 400) could cover just a tiny pieces. I'd like to recommend The Key Stone of European Knowledge to everyone as a reading is worth, but alas, the book is written in Czech and has not been translated to English.

This week I finished reading of "IsGodAMathematician?" and I think I can recommend it as a good enough substitute for the The Key Stone of European Knowledge. It has slightly different focus, it is shorter, yet it covers wider range of topics, yet I believe the way this book describes the beauty of Accient Greek's geometry matches the feel I've got when reading "the keystone" book.

Contents 1 Flaws 1.1 Applied Math 1.2 Aristoteles 1.3 Understanding of God 1.4 The Trust

Flaws

I can only recommend reading IsGodAMathematician. I am especially glad that it references Galileo's thought experiement about speed of falling objects just like Chapter_1 of TheAPIBook. Reading this part was almost like reading my own explanation of the birth of Rationalism. Still I have few comments about differences between the work of Mario Livio and Petr Vopěnka.

Applied Math

At few moments I had a feeling that the description of the history of math is given from the point of view and its applications. Sure, that is expectable when the story is told by a physician, especially if one describes how it is possible math can describe real world. Moreover even the applied point of view is often more complex than what I understand (having master degree from a mathematical faculty - but from computer science department), because the math used these days by physicists is quite advanced.

Still one has to be aware of the limits. For example there is a single paragraph(!) in the whole book dedicated to computer science. One reference to computability theory only! The reach of the IsGodAMathematician is fairly large (thus not everything can be discussed in details), but given the fact that I spent four years at university discussing philosophical aspect of limitations of turing machines, I find omittion of this kind unfortunate. If the book was written for programmers, it would be a huge mistake.

Aristoteles

There is an interesting nuance discussed by Vopěnka in The Key Stone. Vopěnka patiently builds the reader's understanding that there is a significant difference between mathematics as envisioned by Platon and Aristoteles. There is nothing like that in [IsGodAMathematician]]. The whole mathematics inherited from Greeks threated as platonism and Aristoteles contribution is judged as minimal. This is probably acceptable from the physicist point of view, but Vopěnka has to (as a theoretical mathematists and an author of alternative set theory) seek for even the slightest differences. As even slight difference in the initial attitude may have magnificent consequences.

The Platon's geometrical world is given to us and we can just enlight more and more of it by focusing on already existing objects inside it (this is mentioned in IsGodAMathematician as wll). However, according to Vopěnka, Platon's math would be primarily based on evidences - on observing evident truths about the geometrical world. This is a kind of math that never had time to really become wide spread. Why? Because Aristoteles stepped in and gave us logic! Aristoteles changed the Platon's math dramatically by allowing us to use reason and logic (instead of direct evidence) to decide truths about geometrical objects.

The IsGodAMathematician has only small respect to Aristoteles mathematical skills and blames him for making many mistakes (btw. Vopěnka attributes important mistakes to Aristoteles geomatry as well). IsGodAMathematician would rather endorse Platon. But the truth is that the math as we know it (including those who prefer platonism) is significantly influenced by Aristoteles.

Understanding of God

IsGodAMathematician refers to Euclid's Elements a lot. It describes how influencial this book was over centuries, it talks about troubles with the fifth postulate. It gives original as well as modern version of the fifth postulate. IsGodAMathematician clealy explains why Euclid's Elements are so important and influential book for more than twenty centuries. However it fails to mention that (althrough the text of the book remained unchanged), the meaning of the text changed radically.

Vopěnka explains why the original version of fifth postulate does not talk about lines, but only line segments (and why it just extends them, but not indefinitelly). The reason is that accient Greeks were afraid of infinity and were trying to avoid it as much as possible. This has changed somewhere in the Renessaice. Suddenly, instead of requiring a geometer (when looking into the geometric world) to make a line segment twice as long, renaisaince mathematicans rather required to envision line. Reading the Elements and working with infinite lines gives quite a different experience and results in spite the text of the book remained the same.

The explanation of the interpretation shift is also very interesting and has a deep consequences for current math. Vopěnka claims (and I have no reason not to trust) that the mathematicians always invented and use the tools that they could attribute to some imaginable authority. The most skilled authority for Greeks was Zeus. Zeus was the most powerful Greek's god and could definitely make any line segment twice as long. Thus Greek mathematicians safely requested any geomater to be able to extend a line twice. However in case of fifth postulate the number of necessary extensions is not known in advice and depending on how small the angle is it may be very, very high. Even Zeus may be feared to undertake such journey behind the visible horizons (not talking about the case when somebody would request him to do this in hyperbolic space; where he could get lost by traveling to infinity).

Renessance mathematician is different, he has much more skilled garant of operations - the Christian God! He capable of everything, he knows everything, he loves people (including mathematicians), there is nothing that he could not do. How could it come he could not draw a straight, infinite line? Sure he can. As such let's use him as a garant and let use lines in a geometry. Everything becomes so simplified. Actually let start with lines and only derive line segments from them - this is the order how I was tought geometry as a child. The consequences:

1. we have lost the ability to read Euclid's Elements the way Euclid wrote them
2. we have gained enormous ower by having so skilled garant

Thanks to the believe in Christian God the mathematics of renaisance got so dramatic boost. Greeks just could not do it - or they could, but they would consider such behavior rational - they would miss the authority to gurantee it. Interestingly, the understanding of the garant of the mathematical operations was never conscious and over the centuries it vanished almost completely. These days many would deny the necessity of powerful God as a guarant of many mathematical theories (true as mathematics is now disconnected with real world, isn't it?).

Anyway the above leads me to answer to IsGodAMathematician? question: Sure he is, otherwise there would be nobody to draw infinite lines.

The Trust