How is this just a change of basis? To me it seems more like B-theorists have an appropriately coordinate-independent conception of the underlying reality whereas A-theorists are privileging one particular coordinate representation.
In history-of-the-universe space, time-translations are just changes of basis. The difference between A and B is what time you assign to be ‘0’.
When you’re thinking about yourself, it’s appropriate to privilege facts pertaining to yourself. Like, if I’m on a roller-coaster, I will do most of my thinking about accelerations in my personal reference frame. This is a stupid reference frame to use for anything else, even for thinking about the person sitting next to me.
I guess the issue is, it’s easier to think of a block universe in B-theory-mode?
The A-theory and B-theory perspectives are not related by a time translation.
B-theory says that the only objective temporal facts about events in space-time are relative facts. Given two points X and Y, you can ask questions like “Is X in the future relative to Y?” and “What is the space-time interval separating X and Y?” But B-theorists say that it makes no sense to ask questions like “Is point X in the future?” without at least implicitly relativizing the question to some other space-time point or region.
The A-theorist says that there is an absolute answer to the latter sort of question, and it does not depend on any implicit relativization. It is an objective fact about point X whether it is in the future or not. Not whether it is in the future of point Y, not whether it is in the future of the space-time region in which the question is being asked, but simply and non-relatively whether it is in the future. B-theory recognizes no such absolute property of future-ness or past-ness.
I don’t see how this difference can be interpreted as a mere change of basis. I think you are attributing to A-theorists an implicit relativization to the space-time region in which the discussion is taking place; you’re assuming that they have simply chosen a reference frame with a zero in that region, and that all their claims about past and future are actually relative to this frame. But A-theorists are explicit that this is not what they mean. They believe that an event is either in the future or not, and that this is a fact that is independent of reference frame, just like whether X is in the causal past of Y is independent of reference frame.
Like I said in another comment, A-theorists propose radically different ontologies for space and time, and it seems you are under-estimating how radically different these ontologies actually are.
That is trueish, but the point of introducing a distinction between relative facts (before and after) and absolute facts (past and future) is to get a handle in change/becoming,
Note also that presentism is not exactly equivalent to A series.
That is trueish, but the point of introducing a distinction between relative facts (before and after) and absolute facts (past and future) is to get a handle in change/becoming,
This is the stated motivation, although I must confess I have no idea how the A-theory is supposed to be even a partial explanation of becoming.
Note also that presentism is not exactly equivalent to A series.
This is true, but I don’t think I conflated the two in my post. I didn’t say anything about the existence/non-existence of past and future entities or space-time locations. I was talking about the A-theory, not about presentism, although the two are regularly treated as a package deal in contemporary metaphysics.
I actually think the presentism/eternalism distinction is more likely susceptible to shminux’s charge of vacuity than the A-theory/B-theory distinction.
This is the stated motivation, although I must confess I have no idea how the A-theory is supposed to be even a partial explanation of becoming.
I don’t think it’s supposed to be an explanation of becoming, I think it’s supposed to be a model of time that takes becoming into account. Explanations have to ground out somewhere.
I actually think the presentism/eternalism distinction is more likely susceptible to shminux’s charge of vacuity than the A-theory/B-theory distinction.
The opinion has born put forward a number of times, but I am still waiting for someone to substantiate it by putting forward an explanation of how change is equivalent to stasis.
I can see how “time is passing through me” is equivalent to “I am passing through time” ….but those are two dynamic theories.
Do these differnt ontologys have consequences for how we understand physics? For example, suppose society generated a quantum random bit and if it was 1 we allocated all future social resources to pushing Mercury into the sun. If under B theory the future allready exists, then there is a fact of the matter as to whether Mercury is still there. Which then implies that there is a determic result that will happen we we generate the quantum random bit used to make this decision.
I’m no expert on quantum mechanics, but whether or not the result of quantum randomness is in fact determined based on a variable allready in the universe but hidden from us seems like something physics may say something about
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 probably should have used the words “presentism” and “eternalism” instead of “A-theory” and “B-theory”. I’ve added a footnote which I hope clarifies this.
The point is that these are theories about what exists, and not just theories about how we should talk about time. To say that all of time is even “there” to be coordinatized is, in essence, to sign onto the B-theory as opposed to the A-theory.
It seems to me that they’re both perfectly valid and indeed equivalent. Like a change of basis, though, you will seem to have rearranged everything.
ETA: Actually, this IS a change of basis.
How is this just a change of basis? To me it seems more like B-theorists have an appropriately coordinate-independent conception of the underlying reality whereas A-theorists are privileging one particular coordinate representation.
In history-of-the-universe space, time-translations are just changes of basis. The difference between A and B is what time you assign to be ‘0’.
When you’re thinking about yourself, it’s appropriate to privilege facts pertaining to yourself. Like, if I’m on a roller-coaster, I will do most of my thinking about accelerations in my personal reference frame. This is a stupid reference frame to use for anything else, even for thinking about the person sitting next to me.
I guess the issue is, it’s easier to think of a block universe in B-theory-mode?
The A-theory and B-theory perspectives are not related by a time translation.
B-theory says that the only objective temporal facts about events in space-time are relative facts. Given two points X and Y, you can ask questions like “Is X in the future relative to Y?” and “What is the space-time interval separating X and Y?” But B-theorists say that it makes no sense to ask questions like “Is point X in the future?” without at least implicitly relativizing the question to some other space-time point or region.
The A-theorist says that there is an absolute answer to the latter sort of question, and it does not depend on any implicit relativization. It is an objective fact about point X whether it is in the future or not. Not whether it is in the future of point Y, not whether it is in the future of the space-time region in which the question is being asked, but simply and non-relatively whether it is in the future. B-theory recognizes no such absolute property of future-ness or past-ness.
I don’t see how this difference can be interpreted as a mere change of basis. I think you are attributing to A-theorists an implicit relativization to the space-time region in which the discussion is taking place; you’re assuming that they have simply chosen a reference frame with a zero in that region, and that all their claims about past and future are actually relative to this frame. But A-theorists are explicit that this is not what they mean. They believe that an event is either in the future or not, and that this is a fact that is independent of reference frame, just like whether X is in the causal past of Y is independent of reference frame.
Like I said in another comment, A-theorists propose radically different ontologies for space and time, and it seems you are under-estimating how radically different these ontologies actually are.
That is trueish, but the point of introducing a distinction between relative facts (before and after) and absolute facts (past and future) is to get a handle in change/becoming,
Note also that presentism is not exactly equivalent to A series.
This is the stated motivation, although I must confess I have no idea how the A-theory is supposed to be even a partial explanation of becoming.
This is true, but I don’t think I conflated the two in my post. I didn’t say anything about the existence/non-existence of past and future entities or space-time locations. I was talking about the A-theory, not about presentism, although the two are regularly treated as a package deal in contemporary metaphysics.
I actually think the presentism/eternalism distinction is more likely susceptible to shminux’s charge of vacuity than the A-theory/B-theory distinction.
I don’t think it’s supposed to be an explanation of becoming, I think it’s supposed to be a model of time that takes becoming into account. Explanations have to ground out somewhere.
The opinion has born put forward a number of times, but I am still waiting for someone to substantiate it by putting forward an explanation of how change is equivalent to stasis.
I can see how “time is passing through me” is equivalent to “I am passing through time” ….but those are two dynamic theories.
Do these differnt ontologys have consequences for how we understand physics? For example, suppose society generated a quantum random bit and if it was 1 we allocated all future social resources to pushing Mercury into the sun. If under B theory the future allready exists, then there is a fact of the matter as to whether Mercury is still there. Which then implies that there is a determic result that will happen we we generate the quantum random bit used to make this decision.
I’m no expert on quantum mechanics, but whether or not the result of quantum randomness is in fact determined based on a variable allready in the universe but hidden from us seems like something physics may say something about
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.
I probably should have used the words “presentism” and “eternalism” instead of “A-theory” and “B-theory”. I’ve added a footnote which I hope clarifies this.
The point is that these are theories about what exists, and not just theories about how we should talk about time. To say that all of time is even “there” to be coordinatized is, in essence, to sign onto the B-theory as opposed to the A-theory.