Mathematics of physical units and dimensional analysis

Interesting clarifications, Jan.
Regarding agreed upon dimensions, typed algebras, decisions and
adoptions of conventions, and standardization in units in terms of of T
(or 1/T), doesn't all of this also expose why or how, people got/get the
idea that '~consciousness' and/or observation is ~necessary/related in
sorting out quantum mechanical ~results? ...That is, the ~mentalrelated
qualities or aspects are already (subconsciously) inserted in an
earlier, previously adopted set of conventions and thus are already
'nested' in the activity/experience.
In the storyline I advocate and express, 'reality is nested
structured~duality' which means pick a structure and pick a duality
(that is, had I had a better math education, aka, in your terms: 'typed
algebras'). But, with this more unified (NSD) perspective, what we also
have is nested fields within nested fields, rather than just an
idealized or assumed L^3 (lengthcubed) container containing other
collections which somehow, inexplicably pop in and out of
particleantiparticle existence.
One can sort of conceptualize 1/T vibrational features in a nested
fields within nested fields system, perhaps as subdivided tetrahedra
within tetrahedra, whereas it seems a bit more difficult to grasp 1/T
everywhere starting with the initial condition of an unnested L^3,
cubic model.
Thoughts?
Best regards,
Ralph Frost
Reality is nested structured~duality.
On Sunday, July 23, 2017 at 9:41:36 PM UTC4, J. J. Lodder wrote:
> Tom Roberts
>
>> On 7/21/17 7/21/17 7:29 PM, rockbrentwood@... wrote:
>>> Physical quantities form a TYPED ALGEBRA.
>>>
>>> The system of types form an Abelian group with the identity 1
>>> standing for the type of dimensionless quantities
>>>
>>> A type judgement e: T means quantity e has type T, which in dimensional
>>> analysis means [e] = T.
>>>
>>> Addition and subtraction are subject to typerestriction: e + f and
>>> e  f are only defined if e:T and f:T, in which case (e +/ f): T.
>>
>> OK, except there is also the concept of compatible types: cm and inches
>> (in) are not equal (same type), but are compatible:
>> 1cm + 1in = 3.54cm
>> 1in  1cm = .606in
>> 1in / 1cm = 2.54
>> 2 * 1cm = 2cm
>> 1cm / 2 = 0.5cm
>> convert(1in,cm) = 2.54cm
>> convert(1cm,in) = 0.394in
>> ... etc.
>
> That is just saying 'have the same dimension' in other words.
> And that has just the same problem:
> what is, or isn't deemed to be 'compatible'
> depends on your choice of units and dimensions.
>
> For example, is a centimeter compatible with a nanosecond?
> No, of course not, well indoctrinated kiddies will tell you
> with the wisdom of 1793.
> However, these days the nanosecond equals 29.9792458 cm (exactly)
>
> This is what your little GPS unit is doing for you:
> it counts, adds, and multiplies with all those nanoseconds,
> and presents the results of it all in meters or miles.
> (and of course also seconds, for it solves four equations,
> with four equivalent unknowns, to tell you when and where you are)
>
> In a very real sense the metric system came too early.
> (before the relevant physics was well enough understood)
> If we could start all over again
> there wouldn't be a separate length unit at all.
> (and E and B would have the same dimension)
>
> If we ever meet those fabled intelligent LGM in their UFOs
> we may discover that they find our having both meters and seconds
> a very quint idea indeed.
> (just like sane humans find it quaint that those Americans
> have both inches and miles)
>
> Jan
>> On 7/21/17 7/21/17 7:29 PM, rockbrentwood@... wrote:
>>> Physical quantities form a TYPED ALGEBRA.
>>>
>>> The system of types form an Abelian group with the identity 1
>>> standing for the type of dimensionless quantities
>>>
>>> A type judgement e: T means quantity e has type T, which in dimensional
>>> analysis means [e] = T.
>>>
>>> Addition and subtraction are subject to typerestriction: e + f and
>>> e  f are only defined if e:T and f:T, in which case (e +/ f): T.
>>
>> OK, except there is also the concept of compatible types: cm and inches
>> (in) are not equal (same type), but are compatible:
>> 1cm + 1in = 3.54cm
>> 1in  1cm = .606in
>> 1in / 1cm = 2.54
>> 2 * 1cm = 2cm
>> 1cm / 2 = 0.5cm
>> convert(1in,cm) = 2.54cm
>> convert(1cm,in) = 0.394in
>> ... etc.
>
> That is just saying 'have the same dimension' in other words.
> And that has just the same problem:
> what is, or isn't deemed to be 'compatible'
> depends on your choice of units and dimensions.
>
> For example, is a centimeter compatible with a nanosecond?
> No, of course not, well indoctrinated kiddies will tell you
> with the wisdom of 1793.
> However, these days the nanosecond equals 29.9792458 cm (exactly)
>
> This is what your little GPS unit is doing for you:
> it counts, adds, and multiplies with all those nanoseconds,
> and presents the results of it all in meters or miles.
> (and of course also seconds, for it solves four equations,
> with four equivalent unknowns, to tell you when and where you are)
>
> In a very real sense the metric system came too early.
> (before the relevant physics was well enough understood)
> If we could start all over again
> there wouldn't be a separate length unit at all.
> (and E and B would have the same dimension)
>
> If we ever meet those fabled intelligent LGM in their UFOs
> we may discover that they find our having both meters and seconds
> a very quint idea indeed.
> (just like sane humans find it quaint that those Americans
> have both inches and miles)
>
> Jan
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