Is Metric Required for Expansion and Causality in Science?

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Discussion Overview

The discussion explores whether a metric is necessary for concepts of growth, expansion, and causality in scientific contexts. It examines the implications of metrics on the ability to compare states over time and the nature of causal relationships, particularly in relation to discrete and continuous frameworks.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants propose that it is possible to discuss growth and expansion without a metric, citing examples like the cardinality of a set.
  • Others argue that measuring growth inherently requires a metric, as it allows for comparisons between different states over time.
  • One participant questions the meaning of "proceeding in a causal way," suggesting that causality may depend on a continuous connection in time.
  • Another participant asserts that causality does not necessitate continuity, suggesting that continuity is speculative and may be too strong a condition.
  • There is a discussion about whether causality can be derived from topological connections rather than requiring a continuous metric.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of a metric for discussing growth and causality. There is no consensus on whether continuity is required for causality, with some asserting it is essential while others dispute this claim.

Contextual Notes

Participants note the potential ambiguity in definitions of continuity and connectedness, and the implications these have for the discussion of metrics in relation to spacetime and causality.

Mike2
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Is it possible to say that something is growing and expand, or even proceeding in a causal manner, without a metric? Or does expansion require a metric so that it is possible to compare one state of something to its state from a different time?
 
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I think so.

We can talk about size without referring to geometry: we can talk about the cardinality of a set. Of course, I think that has the problem then of how to give a "nice" description of local behaviour. Somehow we need to describe the propagation of the elements of the set in terms of causal laws.

I think quantum causal set theory is related to this problem. Not sure though.
 
Strictly speaking, even "counting" the cardinality of a set is a (discrete) metric.

In order to be able to talk about "growing and expanding" you need to be able to measure and that is what a metric is.

I don't know what you mean by "proceeding in a causal way"!
 
HallsofIvy said:
Strictly speaking, even "counting" the cardinality of a set is a (discrete) metric.

In order to be able to talk about "growing and expanding" you need to be able to measure and that is what a metric is.

I don't know what you mean by "proceeding in a causal way"!
I take it that causality requires time to be continuous so that at "every" step of a sequence of events, no matter how small those steps are, you can say that the past is "connected" to the future. It seems that if time is not continuous or connected, then it is not possible to say that two event in ANY way depend on each other. The causal relation is broken if the time between them is not connected. Same comment for space.

I just am not sure whether an "amount" of spacetime necessarily required a metric. But if it does then causality proves the existence of a continuous spacetime metric. Next, of course, would be to prove the necessity of the equation for the metric.
 
Causality does not require continuity. Continuity is speculation anyway.
 
turin said:
Causality does not require continuity. Continuity is speculation anyway.
OK, continuity may be too strong a word. I think I only meant connected in the topological sense. I'm wondering if we can derive the metric from these kinds of necessities.
 

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