I think that the distinction may be clarified by the mathematical notion of an affine line. I sense that you do not know much modern mathematics, but let me try to clarify the difference between affine and linear space.
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).
The B-theorists are thinking in terms of an affine line. An affine line is somewhat like the A-theoriests real line, but it doesn’t have an origin. Instead, given two points a & b on the affine line, one can take their difference a-b and obtain a point on the real line. The only defined operation is the taking of differences, and the notion of affine line relies on a previously defined notion of real line.
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 sense that you do not know much modern mathematics
… from what do you get this impression, and in what way is it relevant? Yes, there are many parts of modern mathematics I am not familiar with. However, nothing that had come up up to this point was defined in the last 100 years, let alone the last 50.
I have a PhD in physics. I know what an affine space is. If you were thrown off by my uses of basis changes to effect translations, which would signal ignorance since vector addition is not equivalent to change of basis… I did clarify that I was in a function space defined over time, and in the case of function spaces defined over vector fields, translations of the argument of the function are indeed changes of basis.
In physics, we set the origin to be whatever. All the time. This is because we need to do actual arithmetic with actual numbers, and number systems with no 0 are needlessly cumbersome to use. Moving 0 around all the time in context-dependent and arbitrary ways completely defuses the ‘harm’ of A-theory, as far as I can tell.
I apologize for the snipy remark, which was a product of my general frustrations with life at the moment.
I was trying to strongly stress the difference between
(1) an abstract R-torsor (B-theory), and
(2) R viewed as an R-torsor (your patch on A-theory).
Any R-torsor is isomorphic to R viewed as an R-torsor, but that isomorphism is not unique. My understanding is that physicists view such distinctions as useless pendantry, but mathematicians are for better or worse trained to respect them. I do not view an abstract R-torsor as having a basis that can be changed.
I think that the distinction may be clarified by the mathematical notion of an affine line. I sense that you do not know much modern mathematics, but let me try to clarify the difference between affine and linear space.
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).
The B-theorists are thinking in terms of an affine line. An affine line is somewhat like the A-theoriests real line, but it doesn’t have an origin. Instead, given two points a & b on the affine line, one can take their difference a-b and obtain a point on the real line. The only defined operation is the taking of differences, and the notion of affine line relies on a previously defined notion of real line.
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.
… from what do you get this impression, and in what way is it relevant? Yes, there are many parts of modern mathematics I am not familiar with. However, nothing that had come up up to this point was defined in the last 100 years, let alone the last 50.
I have a PhD in physics. I know what an affine space is. If you were thrown off by my uses of basis changes to effect translations, which would signal ignorance since vector addition is not equivalent to change of basis… I did clarify that I was in a function space defined over time, and in the case of function spaces defined over vector fields, translations of the argument of the function are indeed changes of basis.
In physics, we set the origin to be whatever. All the time. This is because we need to do actual arithmetic with actual numbers, and number systems with no 0 are needlessly cumbersome to use. Moving 0 around all the time in context-dependent and arbitrary ways completely defuses the ‘harm’ of A-theory, as far as I can tell.
I apologize for the snipy remark, which was a product of my general frustrations with life at the moment.
I was trying to strongly stress the difference between (1) an abstract R-torsor (B-theory), and (2) R viewed as an R-torsor (your patch on A-theory).
Any R-torsor is isomorphic to R viewed as an R-torsor, but that isomorphism is not unique. My understanding is that physicists view such distinctions as useless pendantry, but mathematicians are for better or worse trained to respect them. I do not view an abstract R-torsor as having a basis that can be changed.
Indeed it wouldn’t. A function space defined on an R-torsor would have a basis which you could change.