Aleph and Irrationality   Leave a comment

I want to share some ideas that I’ve had related to the lost geometrical meaning of old alphabets.

Aleph is the first letter of the Hebrew alphabet. It exists too in other alphabets as the Arabic, Phoenician and Syriac. I’m getting those data from Wikipedia.


Aleph, or Alpha, represents the number one, and as it occurs in the case of numbers with respect to 1, all the other letters of the alphabet are derived from it.

The intuition about letters could have a hidden mathematical meaning is very old. It appeared by example in the development of the so called “Gematria”. It implies to attribute a numerical value to the different letters and to look for the relations between apparently distant words which have the same numerical value.

It looks, it is discussed, the etymological origin of “Gematria” is “Geometry”. I’ve read that other words as “Grammar” could share a similar origin. I will research better on the matter but it looks that those imagined or created or found out mathematical relations between words are limited to the numerical value attributed to those letters.

But mathematics are not a question of abstract numbers and objective quantities related to nothing. They are the constant look for evident or hidden proportionality. The break between arithmetic and geometry appeared since the Pythagorean school was unable to find out a rational and coherent explanation to the discovered disproportionality existent between the legs and the hypotenuse and the apparent proportionality of the square areas related on the Pythagoras’s theorem.

I think there are little details that seem indicate as if the oldest great cultures had had higher mathematical knowledges hidden buy the Greek philosophical schools. Maybe Greek philosophers drank from already dry fountains, saving the last vestiges of those mostly lost cultures.

One of those details is the Aleph.

The symbolic representation of Aleph suggests to me and this is the hypothesis that I would like to research more deeply (feel free to make it by yourself if you want to) that some ancient cultures had a perfectly geometrized alphabet – of which we only have received some barely recognizable echoes – that they knew the explanation about the intrinsic proportionality existent in the Pythagorean theorem, the explanation about how the disproportion created by the displacement of the coordinates which affects to the proportion of the unity itself, the granted of all proportionality, is restored by the compensation of the complementary rational legs and irrational hypotenuses, rational hypotenuses with irrational legs.

In these sense the Aleph – and in a derivative way all the other letters of those ancient alphabets – is not a statical symbol, it expresses the circular movement, the spatial displacement, the changing geometry, and at the same time, the permanency of the unity. The Aleph is a Trinitary symbol which expresses the permanency and the change. It refers to geometry not to mere quantities, not to pure abstractions. It is the expression and guaranty of the beauty of proportionality, and the possibility of its infinite relations.

Maybe the language of birds is a pure geometry too ruled by the similar rules than the inner proportion of the Pythagorean theorem. Maybe intra and extra cellular electrolytic equilibriums are ruled by the same grammar. What is inside is related and complements what is outside.

These phrases can sound as mystic and abstract speculations inside of gnostic traditions or following purely metaphysical intuitions. But they are based simply on a particular comprehension of the pythagorean theorem. A rational one. The one I explained in the previous posts on this blog.


This picture represents the Aleph symbol in the context of the Pythagorean theorem.


The proportionality of those square areas is due to the fact that in all of them we are using rational legs and irrational hypothenuses, and irrational legs and rational hypotenuses, and their total results coincident. If you don’t understand this picture you can try to read the previous post, i explained it clearly.

It is difficult to comprehend that nobody before explored this field in a deep way, that the geometrical perspective of the theory of numbers is unexplored. But the explanation is very easy. The use of irrational numbers without comprehending their existence during almost 2 thousand years, only as a practical and mechanical solution, and later the use of pure mathematical abstractions without reference to geometry or as purely abstract speculations and the statical cartesian methods have avoided that people asked clearly about forgotten questions that seemed already solved in their most elemental aspects.

I think surely we only need to rediscover, one more time, forgotten things.

* I’ve found this interesting book of Arielle Saiber about the geometry of language in Girodano Bruno, ISB-N 0754633217. I expect to get some interesting insights on the matter through its erudite pages:


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