Tuesday, January 26, 2016

Axioms of Torah

Within Judaism, there are matters of belief and practice. For each smallest question, there tend to be many, many answers. However, there are some statements that can be whittled down to the point that there exists only one formulation that is consistent with Judaism - if that statement is rejected, then the whole system fails (such as the 'exactly one God exists' - any negation of that statement is not consistent with Judaism). However, there are other statements on which there exist multiple 'kosher' positions (almost any detail of almost any thing... although some would disagree).

Similarly, as Rav Kook notes in his discussion of Science & Torah, in particular in I:134 of אגרות הראיה (the 2nd of a series of three on this subject) - The Rambam sets out a relatable dichotomy - that Tanach includes language (i.e. metaphors such as 'the hand of God') and laws that are not 'intrinsic' to the nature of God, or the Godly state to which we strive. Rather, these things are included because they are necessary to achieve God's goals - we, mankind, need these things both to uplift ourselves to the level to enact those goals, as well as to be vehicles to deliver these concepts into our minds as they are.

We can imagine alternate scenarios, histories that might have been, where God would have communicated some things differently, and some things different. For example, if our species had come to be with four arms instead of two, or a secondary heart located elsewhere in our bodies, the verses describing the mitzvah of Tefi would have had some differences, to accommodate that different physiology. Had Avraham said "Sarah is my wife" rather than "Sarah is my sister", events might have played our very similarly, other than the details of the words, and perhaps a Divine intervention to save Avraham rather than protect Sarah.


One way to conceptualize this (the difference between 'Truths that must be so' and 'Truths that could be otherwise') is to relate it to undecidable statements in mathematics. For simplicity, I will assume that any particular gestalt of one's position on every matter is reducible to a self-consistent set of axioms.

Then, consider the axioms of Euclid's Elements:

1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Given a line, and a point not on that line, exactly one parallel line exists that intersects that point.

For the first four axioms, if you try to build a framework where one is not true - it does not work, the logic falls apart.

For example, you could change axiom 1 to give two unique lines for every pair of points - this is kind of like saying that 2+2=4, but also that 2+2 = 'other 4', and furthermore, 4 does not equal 'other 4' - neither arithmetic nor algebra can function, as you lose one of the most fundamental properties of all - that a thing is equal to itself!

The fifth axiom (It is often referred to as a 'postulate', for reasons which might interest those who enjoy knowing the history of these things) has alternate formulations that are different - each one contradicts the others, yet each generates its own world.

The Parallel Postulate of Euclid can be true - and you end up with 'regular' straight-line-grid, Euclidean geometry. It is true, and Spartanly elegant.

But - you can also reject it.

You can construct a Geometry (a world-measuring from the original Greek, after all) in which no such point exists - and you have the system of geometry of the surface of a sphere - such as the one on which we live*.

You can declare that there exists no limit to the number of lines which can be drawn through that point - and you have hyperbolic geometry (which is rather harder to simply describe).

Each choice yield a different, and true totality - and the existence of each version does not make the others untrue - they are merely describing different things that share much of their underlying nature (they do share 4 axioms out of five, after all).

Similarly, Judaism has statements which are ironclad, absolute. It also has statements which have many possible iterations - each one true, and each not inimical to the others. And - to only ever have but one of those many is too great a tragedy to bear.

*An oblate spheroid, if you wish to be very מדקדק.

2 comments:

  1. According to Euclid a straight line has no breadth only extension and can be described by points which themselves have no dimension whatsoever- the fundamental datum points to an 'idealized' line which does not exist in nature. This axiom points to the construction of all that follows and we perpetuate for thousands of years the 'falsity' that a 'line' is something we draw in or on some medium. When according to Euclid it simply does not exist! Euclid works because we will be able to establish the language of mathematics through a structure assembled by 'idealized' axioms and be able to construct a self perpetuating 'approximation' validating its own idealization. No one has seen a straight line - how are we indoctrinated to except the shadow for what was for Euclid a truth? We can never have a truth here in this existence because our very 'separateness' and difference precludes us being echad - a one beyond all ones - until we can understand truly what Euclids line 'looks like' we realize how we are compelled to conceptualize and create 'falsities' in order to perpetuate our reason and choice itself.

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  2. Minor point, I dont think the torah has a problem with belief in the existence of many gods, rather, it is the worship of them that is problematic. There are scenes in tehillim and other places where god is administering over the court of the elohim. The commandment against worshiping elohim acheirim also seems to be ambivalent of the existence of other gods. It appears to me that such verses acknowledge the existence of other gods, but damand belief that YKVK rules them all, and that he is the only one to be worshipped.

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