Secular, non-secular mathematics

[viator]

I doubt that people have really thought about mathematics being secular or non-secular, or a numeric value might be secular or non-secular. There is a need to first explain the characteristics of "different forms of mathematics" before we get too deeply involved in the secular issue. I by-passed this issue in the topic "Universe mathematically ordered", as I thought it should be in a different topic.

There is a difference between a "counting system" which is often referred to as the basis of a mathematical system, and formularized mathematics (equations, algorithms, formulas) that use symbols. Symbols representing numbers is a different issue, that is language development.

I would intuitively assume that a "counting system" came first, and it eventually developed to provide for condensed "numbers" to accommodate larger and larger "counts". We are familiar with decimal (base 10) and binary (base 2) methods to accommodate larger and larger counts, but other systems were in use centuries before, i.e, the Babylonian (base-60) and the Mayan (base 20).

http://www.historyworld.net/wrldhis/Pla ... oryid=ab34

They all found methods of manipulating the counting system, producing addition, substraction, multiplication and division processes, and symbols for these process. The "operator symbols" for these basic operations have changed over the centuries but have settled down recently.

http://www.roma.unisa.edu.au/07305/symbols.htm

Somewhere along the line someone applied a "symbol" that was different than the "counting symbol" and the "operator symbol" to represent a specific "numeric value". Thus, those that were familiar with the "symbol" could communicate that meaning without having to recite the number. Different symbols were created to represent other "numeric values" and this lead to the use of formulas to express concepts involving numerical relationships and physical science laws that used symbols exclusively. It was easier to find "relationships" using the symbolic representation of specific values than try to write out the numbers. We current use the small letter c to represent the speed of light, 299792458 meters per second, and this would be laborious to put it in every formula where wanted to use it. Symbols for specific numbers are created on the "fly" in mathematical articles, and I do that also, it is somewhat of a necessity.

It was pointed out to me in the earlier post (and I agree with it) that the symbols that represent numbers are not unit specific, that is you can substitute metric, Imperial or the "unit designators" and their associated numeric values from any other metrology system as long as you are consistent to do this for every symbol that requires unit descriptors. I stated in the "Universe mathematically ordered" topic that it didn't make a difference in many cases what "unit system" was used as the results are not being used to identify the physical laws that govern the universe. When you are trying to identify "universal laws", I stated, 'But when you are trying to develop formulas that will explain the very nature of our physical existence it is desirable to use units that are naturally related to each other and the "system".'

I would like to illustrate my point by stating the length identified as the meter is a secular value. The length that represents the wavelength of the hyperfine transition emission of neutral hydrogen is a non-secular value. The meter was defined by man, the hydrogen wavelength is a part of the universes creation.

We know where the mathematicians got into trouble with the religious authorities, they deviated from using mathematical concepts for commercial purposes by using them to identify characteristics of the universe. The religious authorities (RA) already knew they were the center of the Universe and the only object of Divine existence. The RAs would not tolerate some mathematician coming up with a "natural law" that didn't fit their well established view.

Before we delve deeper into the concept of secular and non-secular mathematics, there is a need to examine some of the philosophical positions regarding mathematics.

The history of how statements of natural laws have come about clearly indicates that all have developed through human mental procedures that include the notion of mathematical abstractions.

http://www.serve.com/herrmann/comparex.htm

Dr. Herrmann's article is more about the process of creating "mathematical abstractions" to fit empirical observations.

The next two articles are related and they are a little easier to read, the Hamming article being a result of the Wigner article.

http://www.lecb.ncifcrf.gov/~toms/Hammi ... nable.html
http://www.dartmouth.edu/~matc/MathDram ... igner.html

There are many other references but I don't think we need to get buried in them to discuss what might make mathematics secular or non-secular. There are references to "secular mathematics" but not to "non-secular mathematics".

I think it would be easy to define non-secular numbers as being something that is inherent in the physical characteristics of the universe. Then, I am sure there are some individuals that will react to the suggestion that a numeric value or numeric relationship (like Pi) is secular or non-secular. I think there is a difference.

[AttentionKMartShoppers]

I doubt that people have really thought about mathematics being secular or non-secular

I have.

[BGoodForGoodSake]

I doubt that people have really thought about mathematics being secular or non-secular

I have.

ROFL!

[angel]

In my view units are never fundamental to an argument. They always are a human conventions (even if they use the hyperfine constant as a base; why not to use two times that? why not the wavelenght the first line of helium spectrum instead of the hydrogen? Why not Planck lenght? ...All these are non-secular lengths)

Moreover, to my understanding physical quantities (I mean the ones which have a value which depends on some unit system) are not numbers but numbers with unit.
You correctly wrote that the speed of light is 299792458 meters per second. That is the same as
299792.458 kilometers per second, something miles per day and so on.
This is due to the fact that kilometers per second=1000 meters per second = something miles per day and so on.

The physical quantity is not to be confused with its numerical value.
The quantity is (or may be) non-secular while the value is always secular.
That is what they wish to catch by using c to denote the speed of light in any unit system. c is not a number it is a speed, i.e. a number with unit.

BTW your difference between secular and non-secular is to my understanding used by physicists since newton or so. Just they call them
conventional or fundamental. That is what they mean when they say that Phanck length is a fundamental constant of nature...

[sandy_mcd]

Angel, nicely and succinctly done.

[viator]

In my view units are never fundamental to an argument. They always are a human conventions ...

Humans cannot communicate numeric quantities or qualities unless they have agreed upon the definitions for units, thus units are very pertinent to an argument.

The fundamental issue in this topic is that I state that a man defined length which has no correlation to a naturally occurring length is a secular length. I don't care what we call the length or what metrology system is used, but the units become a fundamental part of the argument in this topic depending upon whether they are secular or non-secular.

BTW your difference between secular and non-secular is to my understanding used by physicists since newton or so. Just they call them conventional or fundamental. That is what they mean when they say that Phanck length is a fundamental constant of nature...

I have never heard or read of that correlation before. Can you cite some examples where conventional units are referred to as secular units, and fundamental units are referred to as non-secular units?

Can a fundamental unit actually be called non-secular if it uses a unit that is secular?

[angel]

Can you cite some examples where conventional units are referred to as secular units, and fundamental units are referred to as non-secular units?

No I cannot. You are the only one I know using that word (non-secular) to denote fundamental units.
I'm just saying that, from what you write about non-secular and secular units, they seem to be EXACTLY the same thing that physicists call natural (or fundamental) units and conventional units, respectively.

You can look of the use of "fundamental length" on the web to check if it agree with your view.
for example
http://home.comcast.net/~jeffocal/chapter6.htm

There is only one truly fundamental length in nature a length free of all reference to the dimensions and rate of revolution of the planet on which we happen to live, free of any appeal to the complex properties of any solid or gas: free of every reference to the mysterious properties of any elementary particle: what we call today the Planck length,

L=(hG/C^3)1/2= 1.6X10^-33 cm

Can a fundamental unit actually be called non-secular if it uses a unit that is secular?

Sorry, I don't understand the question.

[viator]

Can you cite some examples where conventional units are referred to as secular units, and fundamental units are referred to as non-secular units?

No I cannot. You are the only one I know using that word (non-secular) to denote fundamental units.
I'm just saying that, from what you write about non-secular and secular units, they seem to be EXACTLY the same thing that physicists call natural (or fundamental) units and conventional units, respectively

First, I did not state that the term "non-secular" denoted fundamental units, you created that association.

I do not see how you can conclude that what I refer to a non-secular and secular units seem to be EXACTLY equivalent to the same thing physicists call "natural (or fundamental) units" and "conventional units". My premise for a "unit" or a "numeric" value being secular or non-secular is not based upon the terms conventional or fundamental. If a unit is based upon or derived from an arbitrarily value it is secular.

I don't understand why you would cite a web source that contains such fuzzy thinking.

There is only one truly fundamental length in nature a length free of all reference to the dimensions and rate of revolution of the planet on which we happen to live, free of any appeal to the complex properties of any solid or gas: free of every reference to the mysterious properties of any elementary particle: what we call today the Planck length,

L=(hG/C^3)1/2= 1.6X10^-33 cm

How can one conclude , "...free of all reference to the dimensions and rate of revolution of the planet....", when appended to the numeric value is the symbol "cm". The meter is a clock-dependent operational definition of distance, and the clock "duration of time" is based upon the ephemeris second, which applies to this planet.

I can't understand how the author(s) of that site could make that statement if they even casually examined how the meter is defined. I decided to look into that web site further and in the Introduction, this statement was made:

For example, many people of the fifteenth century believed the earth was flat, even though they could see the circular shadow of the earth moving across the moon during a lunar eclipse.

http://home.comcast.net/~jeffocal/introduction.htm

A flat disk moving between a light source and a display screen will give a round appearance, you cannot assume it is a sphere, you don't have enough information.

The NIST web siter refers to a whole host of numeric values as "fundamental physical constants".

http://physics.nist.gov/cuu/Constants/index.html

It is fairly easy to classify as secular or non-secular many of the numeric values that are classified as "fundamental physical constant". I use the term "many" because I haven't examined all of the constants.

[thechecker]

There is only one truly fundamental length in nature a length free of all reference to the dimensions and rate of revolution of the planet on which we happen to live, free of any appeal to the complex properties of any solid or gas: free of every reference to the mysterious properties of any elementary particle: what we call today the Planck length,

L=(hG/C^3)1/2= 1.6X10^-33 cm

I can't understand how the author(s) of that site could make that statement if they even casually examined how the meter is defined.

As is indicated at

http://www.physlink.com/Education/AskExperts/ae644.cfm

Max Planck himself derived Planck length from the fundamental constant called Planck constant.

[viator]

thechecker, I am not disputing the existence of the Planck length, I questioned the embellishment made by the authors at the ~jeffocal/chapter6.htm site.

... free of all reference to the dimensions and rate of revolution of the planet on which we happen to live, ...

The Planck length is affixed with a metric unit designator, and the meter length is a clock-dependent definition. The duration of the second is based upon the ephemeris second, an earth astronomical value.

The second was defined first and the meter defined in relationship to the duration of the second.

[sandy_mcd]

... free of all reference to the dimensions and rate of revolution of the planet on which we happen to live, ...

The Planck length is affixed with a metric unit designator, and the meter length is a clock-dependent definition. The duration of the second is based upon the ephemeris second, an earth astronomical value.
The second was defined first and the meter defined in relationship to the duration of the second.

Two points:
1) The "Planck length" is independent of the earth. It could be expressed in Planck length units, in which case its value is 1 Plu. One Plu can be converted into cm as ~1.6X10^-33 cm. There is a very important difference between the value of a quantity and the unit it is expressed in. Six of one and half-a-dozen of the other refer to the same number of items. Likewise, the H hyperfine splitting is ~21 cm. Just because it is expressed in earth related units (cm) does not make the hyperfine value dependent on the earth. So it is with the Planck length. The numerical value of the Planck length varies with whatever system of units is used, but the actual length is invariant.
2) The second was defined first and then the meter was defined in relationship to the second and to the speed of light which was assigned an arbitrary value in m/s units. The length of the meter is not determined by the length of the second alone, but also depends on the numerical value given to the speed of light. A second is a unit of time; it is not possible to define a unit of length solely with respect to a unit of time.

[angel]

Dear viator,
it seems the situation is excaping of our control.
I'll try for the last time to clarify the issue.

It seems to me (and I guess to sandy_mcd as well) you are confusing a "quantity" (for example the Planck length) with its "numerical value".
The Planck length is a lenght and as such can be expressed in any length units you please (cm, meter, inches, ...). As any length itself can be used as a length unit, let's call it Pl.

The value of the Planck length is

1 Pl= 1.6x10^-33 cm= 1.6x10^-31 m= 6.3x10^-34 inches=...

Its numerical value changes depending on which length unit you choose.
Of course 1.6x10^-33 and 6.3x10^-34 are different numbers, though I hope we all agree that 1.6x10^-33 cm and 6.3x10^-34 inches are (about) the same length.

When the authours refer to the fact that Planck length is fundamental they are referring to the Planck length, not to its value in cm. As such it has nothing to do with cm, it is very much the same as the fine structure constant you mentioned as an example of non-secular unit.

As I said viator my impression (possibly I'm wrong, but that is my impression) is that you consider 21cm to be secular and 1H (the hyperfine constant) to be non-secular when they are the same length expressed into two different units.
I'm not saying this to prove you are wrong. I'm saying it just because it seems to me you are and sandy_mcd are doing two parrallel discussions which will never meet anywhere.
Of course you are free to make such a difference, though your discussion is in that way loosing any ontological relevance keeping just a lessical meaning.


Among the other funny things is that your definition of "secular"

[If a unit is based upon or derived from an arbitrarily value it is secular. ]

is almost exactly as I would define "conventional quantity".

If this is not enough, I really don't know how to better express it. Sorry for that.

[viator]

There is a very important difference between the value of a quantity and the unit it is expressed in. ... the actual length is invariant.

I agree with that statement completely.

I do not call the actual length of the 1H hyperfine transition emission wavelength secular, it only becomes secular when it is given a numeric value in units that were defined arbitrarily, centimeters or inches as examples. The length is no different if it is 8.3097" or 21.106 cm. The numeric value is secular because it was defined and expressed in a secular unit of measure.

The meter is a secular unit of measure.

If I choose the 1H hyperfine transition emission wavelength as a unit of measure with a value of "1" and its unit designator as the Le, the length of the meter would be 4.7379 Le. In this case a secular number is being expressed using a non-secular unit of measure. It does not make the numeric value non-secular, because we know it is based upon a man defined unit of length, but we will know that the numeric value, as expressed, is related to a real non-secular invariant length.

In the case of the Planck length, it is invariant (we think), but the numeric value was derived and expressed using SI values. The "process" used to define Planck values is quite interesting and it fits into the concept presented in my original post. You might find it interesting, but it will take a little more time to put everything together. I was actually taught how to do that process during my collegiate experience, but I have never used it in a pure form, which Planck did.

[viator]

The initial post on this topic presented some simplistic concepts on how our mathematical system developed. The evolution of the symbolic representation of a numeric value allowed individuals to examine concepts involving physical science relationships that used symbols exclusively and this was extended to dimensional symbols. Planck used the later process but not to everyones satisfaction.

I found a web site that stated why everyone should learn dimensional analysis and that is the first URL in the group below. The second site stated Planck's methodology was a mathematical slight-of-hand. The third site avoided the symbology approach and named the dimensions, and stated that you cannot avoid the final end result simply because of the way you "monkey them" around. The fourth site gives credence to the "dimensional analysis" process but disagrees with the choice of fundamental dimensional units, preferring F (force) over M (mass).

http://www.chemistrycoach.com/use.htm
http://graham.main.nc.us/~bhammel/PHYS/planckunits.htm
http://ebtx.com/ntx/ntx32a.htm
http://www.gatago.com/sci/physics/parti ... 86637.html

It was easy to determine that I am not qualified to discuss the pros or cons of Planck's method after examining the various arguments on both sides. Although I found some web sites with interesting views and conclusions, they will not provide closure on the use of dimensional analysis to identify physical science relationships and associated numeric values.

I found Planck's method somewhat bizarre in my opinion as there is a disconnect between the choice of units and the fundamental characteristics of the physical universe. I would agree with the argument in the last URL above that F (force) is more fundamental than M (mass) because of the arbitrary quality of the kilogram and its stand-alone status in SI.

Among the other funny things is that your definition of "secular"

[If a unit is based upon or derived from an arbitrarily value it is secular. ]

is almost exactly as I would define "conventional quantity".

If a so-called "fundamental qualitity" is derived from arbitrary values, what makes them different from what you would define as a "conventional quality"?

[angel]

If a so-called "fundamental qualitity" is derived from arbitrary values, what makes them different from what you would define as a "conventional quality"?

A fundamental (or natural) quantity is a physical quantity usually derived of fundamental constants of nature (the velocity of light, the Boltsman constant, ...).
A conventional quantity is based upon or derived from an arbitrarily value.

None of the two has anything to do with their numerical value.

Naively one could say that a natural quantity is likely to be defined with the same value by a civilization living wherever in the universe, while there is little chance that a civilization on the other side of the galaxy will find anything cute in a length of 1m.

For example the velocity of light is a peculiar velocity, it is the same whenever and wherever in the universe and it is likely to be known to any civilization in the universe. Hence it is a natural unit for velocities.

Hope it clarifies somehow the issue...