The A-theorists are thinking in terms of a linear space, that is an oriented vector space. To them time is splayed out on a real number line, which has an origin (the present) and an orientation (a preferred future direction).
I think that this analogy is accurate and reveals that A-theorists are attributing additional structure to time, and therefore that they take a hit from Occam’s razor.
However, to be fair, I think that an A-theorist would dispute your analogy. They would deny that time “is” splayed out on a number line, because there is no standpoint from which all of time is anything. Parts of time were one way, and other parts of time will be other ways, but the only part of time that is anything is the present moment.
(I’m again using A-theorist as code from presentist.)
By the way, off-topic, but:
The only defined operation is the taking of differences, and the notion of affine line relies on a previously defined notion of real line.
This is true if affine space is defined as a torsor for the reals as an additive group, but you can also axiomatize the affine line without reference to the reals. It’s not clear to me whether this means that you can construct the affine line in some reasonable sense without reference to the reals. Do you know?
I have always heard the affine line defined as an R-torsor, and never seen an alternative characterization. I don’t know the alternative axiomatization you are referring to. I would be interested to hear it and see if it does not secretly rely on a very similar and simpler axiomatization of (R,+) itself.
I do know how to characterize the affine line as a topological space without reference to the real numbers.
Torsors seem interesting from the point of view of Occam’s razor because they have less structure but take more words to define.
I do know how to characterize the affine line as a topological space without reference to the real numbers.
This is what I was referring to. The axioms of ordered geometry, especially Dedekind’s axiom, give you the topology of the affine line without a distinguished 0, without distinguishing a direction as “positive”, and without the additive structure.
However, in all the ways I know of to construct a structure satisfying these axioms, you first have to construct the rationals as an ordered field, and the result of course is just the reals, so I don’t know of a constructive way to get at the affine line without constructing the reals with all of their additional field structure.
You might be able to do it with some abstract nonsense. I think general machinery will prove that in categories such as that defined in the top answer of
I think that this analogy is accurate and reveals that A-theorists are attributing additional structure to time, and therefore that they take a hit from Occam’s razor.
However, to be fair, I think that an A-theorist would dispute your analogy. They would deny that time “is” splayed out on a number line, because there is no standpoint from which all of time is anything. Parts of time were one way, and other parts of time will be other ways, but the only part of time that is anything is the present moment.
(I’m again using A-theorist as code from presentist.)
By the way, off-topic, but:
This is true if affine space is defined as a torsor for the reals as an additive group, but you can also axiomatize the affine line without reference to the reals. It’s not clear to me whether this means that you can construct the affine line in some reasonable sense without reference to the reals. Do you know?
I have always heard the affine line defined as an R-torsor, and never seen an alternative characterization. I don’t know the alternative axiomatization you are referring to. I would be interested to hear it and see if it does not secretly rely on a very similar and simpler axiomatization of (R,+) itself.
I do know how to characterize the affine line as a topological space without reference to the real numbers.
Torsors seem interesting from the point of view of Occam’s razor because they have less structure but take more words to define.
This is what I was referring to. The axioms of ordered geometry, especially Dedekind’s axiom, give you the topology of the affine line without a distinguished 0, without distinguishing a direction as “positive”, and without the additive structure.
However, in all the ways I know of to construct a structure satisfying these axioms, you first have to construct the rationals as an ordered field, and the result of course is just the reals, so I don’t know of a constructive way to get at the affine line without constructing the reals with all of their additional field structure.
You might be able to do it with some abstract nonsense. I think general machinery will prove that in categories such as that defined in the top answer of
http://mathoverflow.net/questions/92206/what-properties-make-0-1-a-good-candidate-for-defining-fundamental-groups
there are terminal objects. I don’t have time to really think it through though.