In my mathematics degree, dimension simply was the number of elements of a
tuple. The dimensionality of a vector space is the same as the number of
basis vectors needed to get to any point in the space. For example, in 3D
space, one might use a basis (1,0,0), (0,1,0) and (0,0,1) and if these
correspond to the unit vectors in the x, y and z directions the any point
(a,b,c) corresponds to the vector sum a.(1,0,0) + b.(0,1,0) + c.(0,0,1).
Note that a set of basis vectors must be linearly independent, that is, for
a set of vectors V1,...,VN, there does not exist a set of constants
c1,...,ci1,c1+1,...cn such that Vi = c1.V1 + c2.V2 +... + ci1.Vi1 +
ci+1.Vi+1 +...cnVn; All very dull and mathematical. (01)
Among the odd consequences of this definion is that I have a couple of
5dimensional objects in my pockets. When I use a finger to point, it takes
three dimensions to locate my hand and another two to define the direction
in which I'm pointing. Or 6D if you want to model the time when the
pointing is measured. The extrusion of it through time might make it into a
gallery of modern art, though I don't know whether it should make it into an
ontology. (02)
I suspect that calling an ontology "4D" is one of those language games where
a family resemblance is noted to 3D and other notquitesimilar things.
Thomas Aquinas, commenting I think on Avicenna, describes a column of
marching men, each of which sees the person in front and behind, but God,
sitting on the mountain sees the whole column  this was an analogy for
eternity. Perhaps rather than 3D+1 or 4D we could use terms like "on the
road" and "up the mountain" to describe the differing ontologies. (03)
Sean Barker
Bristol, UK
> Original Message
> From: ontologforumbounces@xxxxxxxxxxxxxxxx
> [mailto:ontologforumbounces@xxxxxxxxxxxxxxxx] On Behalf Of Mike
> Bennett
> Sent: 05 February 2011 16:13
> To: doug@xxxxxxxxxx; [ontologforum]
> Subject: Re: [ontologforum] Presentism etc
>
>
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>
> I think you've hit on why I for one find 4D approaches a little
> disquieting, even though I personally see the world in 4+ dimensions
> (hence Hypercube).
>
> I hope it's safe to say that we are all thinking about differences in
> how to represent the world rather than how the world is. In which
> case...
>
> Dimensions (I submit) are that along which we choose to measure
> properties which are orthogonal to any other properties we have thus far
> so measured.
>
> Spatial lengths and distances are measures which can be applied three
> times before they can't be applied again orthogonally.
>
> Time is orthogonal to these, and this is usually the one we choose to
> apply fourth. However in most domains of discourse it presents us with
> the unique feature of causality. This itself presents the problem you
> describe.
>
> In some ways in which we may wish to model the world (though not as yet
> in banking ;) ), time and space can be treated as a single space of
> interchangeable dimensions, giving us 4D spacetime. Though I can't help
> noticing that even black holes have a time before they existed.
>
> Mass remains orthogonal to these (except, again, under certain
> descrptive frameworks which one may wish to ontologize). So that
> requires another dimension along which to measure it. So does electrical
> charge. And so on. Unlike time, there is no consensus about what order
> to label these in. Like time, the question is not whether they are "new"
> dimensions, but whether on certain scales the properties we usually
> measure along them can be regarded as interchangeable within a single 4
> or 5 or 6 dimensional hyperspace.
>
> The 11 dimensional (or is it 7 dimensional?) CalibiYau surface
> described in string theory looks to this naive onlooker like something
> other than a bunch of orthogonal directions along which some properties
> are measured, so I'm curious to find out whether this framework formally
> defines the concept of "dimension" in the same way I have done here. But
> if not, that would not present a challenge for semantically modeling it,
> it just means that this is a different descriptive framework using a
> different definition of "dimension". It seems to me that dimensions are
> sometimes described as actual things rather than measurement constructs,
> when one could as easily say "here is some actual property which is
> orthogonal to the rest" and give the property a name. If we are clear
> about what we mean by "dimension", and in what problem domains we are
> applying that definition, this might be more helpful than all this 3D +1
> minus the square root of whatever.
> Then we have properties of a thing, and dimensions as a property of
> those properties. And interesting constraints or properties about what
> we can measure in some of those dimensions in our domain of discourse.
>
> Coming back to the more immediate space of business problems and
> semantic modeling of them, can't we simply recognise that here are some
> things that have an extent in space and time, along with optionally a
> mass, a charge and so on, and that these properties each present
> particular differences as to how a given property, which occupies a
> given kind of dimension, are modeled. Including for example the unique
> challenges of causality, which means that historical facts can be nailed
> down in a simple space of spatial and temporal facts, while the "future"
> presents unique modeling requirements based on for example how and by
> whom it is predicted. These become problems of provenance.
>
> My favourite practical example about dealing with the future is the
> floating rate note (FRN). This is a bond which pays a rate of interest
> that is pegged to some variable interest rate (say, the London Interbank
> Offer Rate). Suppose this is reset at the end of each month. Then the
> current value of the bond is based on the value of its future cashflows,
> which are a known unknown. Each day, my best guess about the present
> value is based on the future rate of this bond. This is a calculation of
> what the next reset rate would be if it were calculated according to
> today's interest rate, this being the most up to date information I
> have.
>
> Now, for risk management and compliance purposes, I may need to have a
> record of what I perceived the current value of a bond to be at the
> moment I purchased it. If I purchased a FRN a week ago, then I now know
> what it's projected future interest rate (and so current value) is
> today, but I also know what I thought that future value was a week ago.
> These two valuations (past and
> present) are based on the two estimates of the future value of the same
> variable.
>
> That is, there is a time series, in the past, of projected future values
> of the same quantity  the reset rate at the end of this month, along
> with the full set of bond analytics based on current value, that depend
> on this. And these past values matter because they were the basis on
> which investment decisions were made. So there will be data in a system
> somewhere whose meaning we would rather like to ontologize.
>
> If we fail to account for the unique challenges presented by the arrow
> of time, I'm not sure we would be able to properly model these sorts of
> scenarios. Time is not just a dimension, it's a dimension along which
> every point in the past possesses an infinite number of unknowns about
> its future. We have a future, but every past has a future of its own.
>
> Mike
> (04)
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