Tuesday, August 27, 2013

Mathematics as Pure Relation

It seems to me that the difference between mathematics and other studies is that mathematics consists entirely of relations.

Whereas other studies consist of objects and their relations to properties or their relations to one another (for example, in biology we say that the platypus is a mammal that lays eggs, which could be cast as a statement about how a platypus relates to the mammal property and the egg laying property; in history we say that in 49 B.C. Julius Caesar crossed the Rubicon - which can be cast as a spatial relation between Caesar and the Rubicon and a temporal relation between 49 B.C. and all other dates), mathematics can be seen as being nothing but relation. What is the definition of '5?' We could define it as a relation - it is the relation of being less than 6 and more than 4. This is rough, of course, if one considers the infinite fractions or decimals that could also fit that definition, but what else is a number save a relation to all other numbers?

Mathematics can be seen as the study of the continuum relation. Can a relation exist apart from objects? Are there relations simply relating [void] to [void]? At this time I see no need to conceive of relations in this way. My concept of relations requires that things be related to one another.

5 + 5 = 10 is an abstracted mathematical relation. It is true in the way that any concept can be true: it is coherent, it does not violate the principles involved in its own definition, but it is not yet bound to the world of phenomena. Five marbles thrown into a jar of five marbles will produce what we call ten marbles in a jar; the abstract logical construct has now been bound to the world of phenomena.

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