[from CanAnArchitectureEmerge]

As a RelationalWeenie, it is my view that it is very unlikely that "relational math" could "organically" or incrementally emerge through XP-like processes. If left to organic growth, you tend to end up with NavigationalDatabase-like structures, which I don't like, except maybe on a small scale. (See OoLacksMathArgument). Further, one would tend toward ReinventingTheDatabaseInApplication.

''As a ChaoticWeenie, it is my view that all math organically and incrementally emerged through evolutionary processes. All ideas (and brains) grow organically.''

Do you consider math something that is invented or discovered?

[Is this a trick question? Of course math is discovered, since it is the language of explaining an already existing entity - that is, the Cosmos. It is not "invented," per se, but math's expression and symbology are created to match the existing pattern.]

''Is this a trick question? Of course math is invented like any other language. We didn't find English lying under a rock, nor did we find mathematics.''

[Hang on a sec - the question was about ''math'' per se, not about the ''expression'' of math. The nature of the physical relationships between numbers (two added to two still yields four, no matter how you express the relationship) always holds, independent of time and physical existence. The field of mathematics is an attempt to discover these relationships and express them in terminology that we do invent on the fly.]

''Don't eat the menu. Mathematics is a set of languages that describe some real things and some imaginary things. There are numbers and relationships described by mathematics that are highly unlikely to occur anywhere outside a primate nervous system. The real world seems to be quantized in all dimensions (space, time, energy, matter, gravity, etc.), yet mathematics easily models continuums.''

The real world doesn't look quantized in all dimensions. I am not aware of a decent model of quantum physics which uses quantized space and time, except as approximations. Energy is quantized under certain circumstances, but the quanta don't line up, so there is no base unit of energy that everything is restricted to.

''All models built on Planck's constant imply quanta for time and space. The smallest unit of energy defines the shortest event duration. Multiply by the speed of light and obtain the smallest unit of distance. Energy is quantized under all circumstances.''

[''Concur. Mathematics is a pure theoretical science. There is no such thing as "2" and outside of mathematics "2" is an adjective not a noun, i.e., it cannot be acted upon a verb like plus. Addition, subtraction, multiplication, division, etc., are man-made concepts that we use to model the real world; they do not exist in the real world.'']

Nonsense. The "twoness" of the of the sun and moon, the "threeness" of the sun, moon and earth, etc., were there (albeit undescribed) before man existed, so we certainly didn't invent them, just the means of studying them.

''Nonsense yourself. The sun and moon have no more "twoness" than the sun and a booger. We categorize them as "circles we can see in the sky" and say they are two members of that category, but we created the category and we put them in it.''

Nonsense again. Booger would do fine. No more "twoness", no less "twoness", but the same "twoness". It's only the quantity that counts, not what they are, nor whether the combination has been thought of by someone.

''But quantity depends on labels we create and apply.  Every booger is unique, yet we're able to ignore their uniqueness and pretend they belong to a set.  Without set membership everything's unique, "oneness".  There is no essential "twoness" locked inside the sun and every booger, or inside every booger pair, or inside every possible pairing of combinations of particles, waiting for a mathematician to release it.  Mathematicians assign "twoness" to sets they dream up.''
----

Relational did evolve. De facto. It's expression was simple enough, use associative lookup instead of location lookup. File 'names' were doing this already, it just needed applying in a more general sense of querying. All the relational math came along after the fact, as usual.