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The philosophical 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 its reading is worth it, 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 Key Stone of European Knowledge. It has slightly different focus, it is shorter, it covers wider range of topics, yet I believe the way this book describes the beauty of Ancient Greek's geometry matches the feel I've got when reading "the keystone" book.



I can only recommend reading IsGodAMathematician. I am especially glad that it references Galileo's thought experiment about acceleration 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 of 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 the 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 sentence 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 aspects and limitations of Turing machines, I find omission of this kind unfortunate. If the book was written for programmers, it would be a huge mistake.


There is an interesting nuance discussed by Vopěnka in The Key Stone of European Knowledge. 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 is categoried 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 mathematicians and author of alternative set theory) seek for even the slightest differences. As even a minimal difference in the initial attitude may have magnificent consequences.

The Platon's geometrical world is given to us and we can just enlighten more and more of it by focusing on already existing objects inside it (this is mentioned in IsGodAMathematician as well). 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 geometry 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 influential 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 clearly explains why Euclid's Elements are so important and influential for more than twenty centuries. However it fails to mention (although the text of Euclid's book remained unchanged) that 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 indefinitely). The reason is that ancient Greeks were afraid of infinity and were trying to avoid it as much as possible. This has changed somewhere in the Renaissance. Suddenly, instead of requiring a geometer (when looking into the geometric world) to make a line segment twice as long, Renaissance mathematicians rather requested to envision whole line. Reading the Elements and working with infinite lines (instead of only line segments) gives quite a different experience and feeling 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 used only 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 geometer to be able to extend a line twice (at least in a mind). However in case of fifth postulate the number of necessary extensions is not known in advance. It depends on how small the angle is and 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).

Renaissance mathematician is different, he has much more skilled guarantor of operations - the Christian God! Such God is capable of everything, knows everything, 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 guarantor of our mind operation and let's use infinite 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 taught 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 power by having so skilled guarantor
  3. we are now fearlessly crossing our horizons thinking about infinity as it would be in our reach

Thanks to the believe in Christian God the mathematics of renaissance got so dramatic boost. Greeks just could not do it - or they could, but they would not consider such behavior rational - they would miss the authority to guarantee it.

Interestingly, the understanding of the guarantor 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 guarantor of many mathematical theories (true as geometry is now disconnected from real world as IsGodAMathematician puts it as well).

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

The Trust

There is last thing to add to IsGodAMathematician which is present in The Key Stone of European Knowledge book. IsGodAMathematician explains why today, we no longer need the Christian God for any mathematical theories. We understand there are many geometries, we don't know which one applies to the real world, we don't care. However we have not isolated ourself from the God's influence completely - there is a significant area of applied mathematics that is built around deep trust to Christian God - physics!

Whenever we apply or verify our knowledge we can do it only inside boundaries of the known world. What is behind a horizon is unknown. Old Greeks would think twice before crossing their horizon. Not only the danger there may be n-times as big, you may even hit an Apeiron as Vopěnka nicely explains. We have no such fear. Rather we believe that the same laws that can be applied here can be applied behind horizon as well. No surprise we often face paradoxes!

We have learned that we cannot apply Newton's physics to something fast. We know we cannot apply it to things too small. Greeks would give up can admit that they cannot estimate what is behind the horizon (of something too distant or something too small). What is our response? We still believe that everywhere in the cosmos the same physic laws can be applied. We still believe we can explain them once all. Why? Yes, the enormous success of science in last few centuries might give us some trust. But deeper below that is the old good Renaissance trust to the biggest guarantor. The Christian God is good, loves scientists and does not play dice, or does it?


In spite, or maybe even because the flaws I listed above I believe it is worth to read the IsGodAMathematician book. I enjoyed it. The book gives a clear and wide overview of the history of the math. It describes important milestones on its evolution paths. If you want to ask additional questions that Petr Vopěnka answers in The Key Stone of European Knowledge and which I outlined above, you may treat the IsGodAMathematician book as a gentle introduction to the topic. Then you can either answer them yourself or learn Czech and read the original.

Even if you are programmer you may find IsGodAMathematician an interesting read. In spite the book ignores any achievements in computer science, it will give you excellent insight into rationalism and empiricism which you can later (after reading TheAPIBook) use to help your users to become clueless. Because your fellow developers don't care whether God (e.g you as author of the API) plays dimes or not, rather whether he versions the world properly.


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