Evolution of the concept of axiom: from self-evident to inexplicable in principle

Y. B. Karasik, Thoughts Guiding Systems Corp., Ottawa, Canada. e-mail:karasik@sympatico.ca

The Greeks came up with a great idea: derive not obvious facts from self-evident ones. They called the latter axioms and successfully implemented this program in geometry. Euclid's "Elements" had shown how to derive all not obvious geometric knowledge from a small set of seemingly self-evident facts.

For thousands years after Euclid people were trying to repeat his feat beyond mathematics. But to no avail. It was hard to identify self-evident facts in other areas, such as mechanics, for instance. Those who tried to failed to explain mechanical phenomena from "self-evident" facts. Gradually the idea that fundamental facts in other areas do not have to be self-evident was adopted. These not self-evident fundamental facts were called "laws of nature". And all other facts started to be derived from the not-obvious laws. The attempts to derive laws from something self-evident ceased.

Then Gauss, Bolyai and Lobachevsky dealt the beautiful idea of the Greeks a mortal blow: they came to a conclusion that Euclid's axioms are not self-evident but just ones of several possible. Since then the notion of axiom lost its lustre. It was no longer self-evident but just something elementary. The great Greek program of deriving all not obvious phenomena from the obvious facts was replaced by a mediocre program of building complex structures from simple blocks.

But simple blocks have that drawback that you always can find something simpler, whereas you cannot find anything more obvious than self-evident. That is why new, more fundamental laws of nature, get discovered from time to time, and the old laws get derived from the new ones.

To return lustre to the idea of axiom one can put into it the following content: axiom is not something self-evident or elementary block, but something inexplicable in principle.

• Definition: A phenomenon (except language) is called inexplicable in principle if any its explanation uses this very phenomenon.
For example, the phenomenon that there exist distinct objects is inexplicable in principle: any its explanation would use distinct words. Similarly, existence of order is also inexplicable in principle: any its explanation would use the order of words. Etc.

Can such inexplicable in principle phenomena be put into the foundation of the human knowledge ? Could it be that all our knowledge is built of them ? Ultimately a child starts cognizing the world by first learning to recognize objects (i.e. by learning that objects are distinct). In other words, cognition begins with structuring the unstructured chaos. Then he learns the idea of order, etc. Then he builds upon these ideas and acquires more complex knowledge.

If we could reproduce this process in all detail then we would be able to derive all human knowledge from the inexplicable in principle facts. Is it not a decent substitute for the program of the Greeks ?