Mathematical constant or physical constant

[FrankM]

I was told that a fundamental mathematical constant and a fundamental physical constant can never be the same. The original discussion involved 2pi and the sqrt(2). Pi is understood to be a mathematical constant. What happens when pi is multiplied by 2? Is that numeric value always a mathematical constant or does it sometimes describe some fundamental characteristic of the physical universe?

 

The simple act of adding a dimension to a mathematical constant (i.e. 1), transforms it to something else, and I wonder if this always holds true with 2pi. Is 2pi sometimes used to imply angular characteristics but not dimensioned with angular units? The sqrt(2) is a mathematical constant that is associated with a geometric relationship that applies to a specific angle. Even though it is always associated with a specific angle I have never seen it dimensioned with an angle unit.

 

Can a mathematical constant ever have a dual identity as a physical constant?

[CraigD]

Can a mathematical constant ever have a dual identity as a physical constant?

Yes, I think, in a practical sense. However, I think the question hints at a fundamental confusion concerning the deep meaning of these 2 terms.

 

It’s necessary with this question to have definitions of the terms “mathematical constant” and “physical constant”, so I’ll state some. My definitions will not agree with every existing use of the terms, but will serve, I hope, to communicate the ideas I wish to express.

 

A mathematical constant is a number represented exactly using only integers and arithmetic, including “procedural” operations, such as summation. [math]1[/math], [math]\frac{-2}3[/math], [math]42660[/math], [math]\pi[/math], [math]sqrt{2}[/math], [math]\sqrt[3]{3}[/math], and [math]sqrt{-1}[/math] are all examples of mathematical constants, of which there are an infinite number.

 

The representation of a mathematical constant is not unique. For example, [math]42660[/math] could be written [math]43423-763[/math], or an infinite number of other ways.

 

Of these, all but [math]\pi[/math] are actually written using only integers and arithmetic. Because [math]\pi[/math] is a frequently used constant, and because writing it using only intergers and arithmetic takes an inconvenient amount of space, (eg: [math] 4 \sum_{n=0} \frac{(-1)^{n}}{2n+1}[/math])it’s usually referred to be this special symbol.

 

Most mathematical constant aren’t useful enough to have special names or symbols. Of the above, only [math]1[/math], [math]sqrt{-1}[/math], and [math]\pi[/math] are commonly considered “special” enough for this distinction.

 

A physical constant is a measurement of one or more physical qualities, optionally subjected to arithmetic. A physical constant may be in terms of some unit – for example, 182 cm (my height) or 299792458 m/s (the speed of light in vacuum) - or may not – for example, 3.15, (the ratio of the length of the longest and shortest finger bones of my left hand), or 0.007297352537650, the fine-structure constant. As with mathematical constants, some physical constants are more important than others, though it depends on who is interested – my tailor, for example, likely considers my height of more importance than the fine structure constant. The usefulness of a physical constant is often related to how little it appears to change, or is believed to have changed or be likely to change.

 

The ratio of the length of the shortest string that encircles without overlapping an accurately machined cylinder © to the longest one that can be stretched across its flat end (d) is [math]\frac{c}{d}[/math], and is, if the cylinder is machined, conserved, and measured with sufficient accuracy, for practical purposes, the same number as the mathematical constant [math]\pi[/math]. Few would disagree with the claim [math]\frac{c}{d} = \pi[/math].

 

[math]\frac{c}{d}[/math] and [math]\pi[/math] are not, however, the same number, nor the same kind of thing, because of how I have defined. [math]\pi[/math] can be perfectly represented by an arrangement of discrete symbols, such as summation expression given above. [math]\frac{c}{d}[/math] is doomed to be forever a work in progress, and due to the finite and imperfect nature of physical measurement, can never be perfectly accurate. In a very important sense, [math]\pi[/math] contains much more information than [math]\frac{c}{d}[/math], because it can supply an infinite, completely well-determined sequence of numerals, while [math]\frac{c}{d}[/math] contains only as much well-determined data as the precision of the measuring process was able to supply.

 

One might philosophically call the distinction between mathematical and physical constants the distinction between the ideal and the experiencable, or cynically call it a matter of semantic.

[FrankM]

A mathematical constant is a number represented exactly using only integers and arithmetic, ..... of which there are an infinite number.

 

By definition one can have an infinite number of mathematical constants, but a fundamental mathematical constant has an intrinsic character that is unique. 1 can be called a fundamental mathematical constant, but any number that is a multiple of that number does not make it unique. Some numbers appear to be unique, 299792548, but only if we assign units to them. It is simple expediency to use [math]\pi[/math] instead of c/d, as the result is always one particular value. Since c/d is reflective of dimensions in the real physical world it is not too much of a stretch to consider [math]\pi[/math] a fundamental physical constant as well as a fundamental mathematical constant.

 

Multiplying 1 by 2 creates a another constant, just larger than 1 but not particularly unique. But it seems multiplying [math]\pi[/math] by 2 is another story. I see 2[math]\pi[/math] and other multiples of [math]\pi[/math] in a lot of formulas that eventually result in what are called fundamental physical constants. 2[math]\pi[/math] is often used in its angular sense and in an equation I can't tell the difference between plain 2[math]\pi[/math] or the angular 2[math]\pi[/math].

 

[math]\sqrt{2}[/math] is a pure mathematical quantity but in geometry it is a fundamental value related to a particular geometric configuration. Does this make that value a fundamental mathematical constant in that use only or in all uses?

 

Are there any instances where multiplying together two fundamental mathematical constants will result in a fundamental physical constant?

[FrankM]

I did not mention in the previous post that 2[math]\pi[/math] has a particular duality, as that numeric value can represent either a frequency or an angular frequency. The duality of this convergence point is useful when representing wavelengths and frequencies in a geometric form.

 

http://vip.ocsnet.net/~ancient/PrimitiveWF.pdf

[CraigD]

By definition one can have an infinite number of mathematical constants, but a fundamental mathematical constant has an intrinsic character that is unique. 1 can be called a fundamental mathematical constant, but any number that is a multiple of that number does not make it unique.

…

Are there any instances where multiplying together two fundamental mathematical constants will result in a fundamental physical constant?

I can’t meaningfully comment or address this question without a formal mathematical definition, or one that can obviously be formulated in formal terms - of the term “fundamental mathematical constant”.

 

This definition, for example, is obviously formalizable:

A mathematical constant is a number represented exactly using only integers and arithmetic, including “procedural” operations, such as summation.

 

The hint that a “fundamental mathematical constant” “has an intrinsic character that is unique” doesn’t give a reader familiar with formal methods – at least not this one - enough to go one to put it into the formal terms necessary to get to the heart of the mathematical subject under discussion. FrankM, can you provide more defining information, or, better yet, formalize the definition of “fundamental mathematical constant” yourself?

Multiplying 1 by 2 creates a another constant, just larger than 1 but not particularly unique. But it seems multiplying [math]\pi[/math] by 2 is another story. I see 2[math]\pi[/math] and other multiples of [math]\pi[/math] in a lot of formulas that eventually result in what are called fundamental physical constants. 2[math]\pi[/math] is often used in its angular sense and in an equation I can't tell the difference between plain 2[math]\pi[/math] or the angular 2[math]\pi[/math].

This reminds me of Bob Palais’s interesting opinion column in Mathematical Intelligencer, which is linked to and discussed in the 11138 thread.

 

Personally, I’m happy with the conventional meaning of [math]\Pi[/math], but there are sensible arguments that life would be easier if it were some multiple, or even some fraction, of its conventional (3.14 …) value.

[FrankM]

This reminds me of Bob Palais's interesting opinion column in Mathematical Intelligencer, which is linked to and discussed in the Pi is Wrong! thread.

 

Pi Is Wrong!

 

I can understand why Bob Palais wrote the article the way he did, the comments in the "Pi is Wrong!" thread illustrate a fraction what he had already experienced.

 

"When I have suggested to people that [math]\pi[/math] has a flaw, their reactions range from surprise, amusement and agreement, to "Of course, I knew it all along," to dismissal, to indignation."

 

The article was published in 2001. When I began to read the article, I was hoping he would distinguish between 2 times the Archimedes' Constant, [math]\pi[/math], which has no dimensional notation, and that of 2[math]\pi[/math] that has the dimensional notation of frequency, cycle/T, and that of 2[math]\pi[/math] that has the dimensional notation of angular frequency, rad/T. In the article he stated the "newpi" symbol he used is formed by two [math]\pi[/math]'s squeezed together so that a leg of each overlap. I thought his symbology was quite appropriate. Palais did not give a name to his new symbol, but I will refer to it as double-pi.

 

lefthttp://hypography.com/forums/attachment.php?attachmentid=1582&stc=1&d=1183049004[/img]

 

When Palais used Dirac's Constant as one of his examples, it illustrated why there is a need to replace the current symbology, as it does not distinguish between one of the three 2[math]\pi[/math] forms that can occur in an equation. Below the equation for Dirac's Constant he provides an equation for T, where double-pi is divided by [math]\omega[/math], which seems to indicate that his symbol for 2[math]\pi[/math] can represent either frequency or angular frequency.

 

When I illustrated the use of 2[math]\pi[/math] as legs of a right triangle, I stated that the numeric value of 6.2831 (2[math]\pi[/math]) can represent both a frequency and angular frequency. I never suggested it might mean 2 times [math]\pi[/math], the ratio, but 2[math]\pi[/math] is ambiguous.

 

http://vip.ocsnet.net/~ancient/PrimitiveWF.pdf

 

This is exactly the problem that can occur when 2[math]\pi[/math] appears in an equation, it might actually mean 2 times[math]\pi[/math] the ratio, or 2[math]\pi[/math] the frequency or 2[math]\pi[/math] the angular frequency.

 

In another of his articles Palais used his double-pi symbol extensively.

 

The Natural Cosine and Sine Curves

 

Quite far down in the article he identified how his father created a "Tex" macro to generate the symbol.

 

It is my contention that 2[math]\pi[/math] can represent both frequency and angular frequency simultaneously, which is why the relationships shown in PrimitiveWF.pdf are valid.

 

When the numeric value, represented by 2[math]\pi[/math], appears in Dirac's Constant, does it represent a characteristic of the physical universe or is it just 2 times [math]\pi[/math], a mathematical value? Does it have a dimensional notation?

[FrankM]

Bob Palais did put a name on the new symbol for 2[math]\pi[/math], he called it newpi. He had discussed the issue with the Unicode people.

 

Does anybody know whether the 2[math]\pi[/math] used in Dirac's Constant was used in the angular sense or as the ratio?