Consideration of the difference between the use of the term 'space' in Physics and Mathematics.

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Posted (edited)

Again and again members ask the question "what is space ?"

Indeed we have at least currently active threads which include discussion of this question.

So what do members consider the difference between the two uses to be ?

Edited by studiot
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Posted (edited)

I am not sure what its scope was in psysics , but in maths,

a space is that a set and with  at least a metric defined on this set.

Edited by ahmet
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27 minutes ago, ahmet said:

I am not sure what its scope was in psysics , but in maths,

a space is that a set and with  at least a metric defined on this set.

I think you have captured the spirit of what I consider to be the difference but your example is not necessarily a mathematical space.

Consider the set {AB, BA}

What metric would you offer on this set ?

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Posted (edited)
41 minutes ago, studiot said:

What metric would you offer on this set ?

any metric should satisfy these criteria / conditions:

1) For every x,y

$d(x,y) \geq 0; if, d(x,y)= 0 --> x=y$

2) $d(x,y)=d(y,x)$

3) $d(x,y) \leq d(x,z)+d(y,z)$

Edited by ahmet
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1 minute ago, ahmet said:

any metric should satisfy these criteria / conditions:

1) For every x,y

d(x,y)0;if,d(x,y)=0>x=y

2) d(x,y)=d(y,x)

3) d(x,y)d(x,z)+d(y,z)

So what metric would you offer that satisfies your conditions ?

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4 minutes ago, studiot said:

So what metric would you offer that satisfies your conditions ?

I do not know any metric for this set,specifically.

but have you understood my wordings as stating;

we could define a metric at every sets?

if so, I did not mean that thing. sorry for mistelling if it is being understood so.

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Posted (edited)
4 minutes ago, ahmet said:

I do not know any metric for this set,specifically.

but have you understood my wordings as stating;

we could define a metric at every sets?

if so, I did not mean that thing. sorry for mistelling if it is being understood so.

No that is not what I understood you to have meant.

I understood you to have meant that in order to have a mathematical space you must have both a set and also a metric specified on it (and perhaps more besides)

So my comment about the spirit of the mathematical space referred to this.

So I offered you a set, which is the space of positions two (a pair of) dance partners may take up  and asked for a metric for this space.

Edited by studiot
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We have vector spaces also. But..

Presumably some more and relevant descriptions available. But if you name it as "space",then to my knowledge, the appearing explanations should be valid.

Note please there are many operations and broad content in spaces  (and there are many spaces such as metric, normed, inner product spaces and specific examples to these)

Metric spaces are the most discrete or narrow spaces as i know.

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Posted (edited)
6 hours ago, ahmet said:

We have vector spaces also. But..

Presumably some more and relevant descriptions available. But if you name it as "space",then to my knowledge, the appearing explanations should be valid.

Note please there are many operations and broad content in spaces  (and there are many spaces such as metric, normed, inner product spaces and specific examples to these)

Metric spaces are the most discrete or narrow spaces as i know.

OK I agree with this but consider.

What do you need to have as a minimum for a vector space ?

1) Well you need a set of vectors.

2) You need a set of rules (the vector space axioms, which are really rules of combination of the vectors and elements of the field set)

3) You need a set of objects that form an algebraic field

4) You need a set of permissible operations.

Now for the minimum (4) is combined into 2 as the one and only operation specified.

When more operations are specified then more rules are required and more structure is available, using member set (4). You may also then need a set of definitions.

So a vector space is a set that contains at minimum 3 sets (not subsets) as members.

Note that in general the rules in the rule set do not apply to the 'space' itself.

Therein lies one difference from Physics.

Edited by studiot
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In mathematics I know of two ways in which you can define something that merits the name and notion of space. One is that based on a metric. A metric allows you to define a distance, and from there a notion of open sets. This is called a metric space, which is the one @ahmet is talking about.

Also in mathematics, you have the notion of a topological space. You give structure to this entity without even having a concept of distance, but only the notion of inclusion, intersection, etc. upon which you define the notion of base of open sets. I remember this definition, but I must confess I have no use for it, or know how to handle it efficiently.

As to physics, I don't have a complete answer for you, but I think it is worth noting that some physicists today are wondering whether space is an emergent property, or epi-phenomenon; that quantum fields perhaps (the stuff), are the really basic concept, and space & time are some kind of derived property. That quantum fields somehow "generate" extension, would be a way to put it.

I think Einstein had a very similar notion, and that he thought that space-time is derived from relationships between matter and radiation, which are the actual be-all, end-all of all that we perceive. But maybe I misunderstood.

I think it's entirely possible that conscience produces some constriction that doesn't allow us to perceive the ultimate nature of space-time and their contents as they are, and we are cursed by our very own nature of physical systems that project the world around them in the way of a 3+1-dimensional map, but the real business of what's going on is hidden safely beyond what we perceive and, perhaps (I hope not), what we can perceive, as J. B. S. Haldane put it:

Quote

My own suspicion is that the universe is not only queerer than we suppose, but queerer than we can suppose.

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On 5/18/2021 at 6:40 AM, ahmet said:

a space is that a set and with  at least a metric defined on this set.

A topological space does not necessarily have a metric. The definition of space is far more general.

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22 minutes ago, wtf said:

A topological space does not necessarily have a metric. The definition of space is far more general.

yes , thanks for reminding.

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We never "bump into" space do we?

Isn't that a handy way to understand whether something we observe has an objective existence or whether it is a reflection of our thoughts?

We can change our thoughts (as when we dream)  but when we "bump into"  external circumstances we quickly learn on which side our bread is buttered and align them accordingly(though some of us are slow learners

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6 hours ago, geordief said:

some of us are slow learners

for example ,like you

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Just now, ahmet said:

for example ,like you

Which is what I was implying.

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Don't remember much about mathematical spaces, but there are several differing examples of physical spaces also.

You can define space as a certain height above Earth's atmosphere, and even then there are several qualifiers for the space between planets, stars and galaxies, as interplanetary, interstellar, or intergalactic space.
The space where all possible states of a system are represented as unique points, is called a phase space.
And, of course, the space represented by X, Y, and Z co-ordinates, once a co-ordinate system ( cartesian in this case, but there are many others ) is assigned to a volume.

I'm sure there are many more, and I don't really have a problem with the different definitions.
If one qualifies which definition they are using, even a newly made-up one, so that we are all on the same page, we can engage in a fruitful discussion.
Often many non-math or non-science noobs will use a wrong definition, and they are often asked what they mean by such a term, as it is non-standard.

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