Nereid

2006-Mar-30, 09:32 PM

In the why math? thread (http://www.bautforum.com/showthread.php?t=39653) in the ATM section, I said I'd start a thread (http://www.bautforum.com/showpost.php?p=711731&postcount=56), here in General Science, on infinities in your daily life. This is it.

Let's start with temperature.

We all know that F(ahrenheit) and C(elsius) are quite arbitrary - both for the zero point and the scale. Along came K(elvin), which at least has a non-arbitrary zero point, even if its scale is still arbitrary.

We've no doubt read about how close to 0K researchers have been able to get, and maybe we've read about a way to 'show' how unattainable 0K is, by taking the log of the temperature (0K then becomes -∞, which is, 'of course', unattainable).

But why do we use a linear scale for temperature? Why don't we use a logarithmic one? After all, such a scale would 'show' the unattainability of '0K' much more clearly, wouldn't it? And anyway, we already have several logarithmic thingies, don't we? pH, and decibels, and earthquakes (think Richter), for example.

It turns out that there is an interesting aspect to temperature and logs ... and it has to do with entropy. In one definition of temperature, it is just dE/dS, where E is the total energy of the system, and S the entropy; and S is just the logarithm of the number of microstates of the system.

And this definition leads to the possibility of negative temperatures (http://en.wikipedia.org/wiki/Negative_temperature), which, despite what you might think, are not colder than 0K, but hotter than +∞!

So, where might you encounter an infinity (temperature in this case) in your daily life? The CD/DVD player in your PC likely has a component that goes through a state which has a temperature of +∞ (and one with a temperature of -∞) during its normal operation. So does the barcode scanner in your local supermarket. And a great deal of other circumstances too.

But this is just a mathematician's (or physicist's) trick, right? Nothing is really infinite in nature, right?

Well, I guess that's up to you to decide - how does 'infinity', a mathematical concept, relate to 'reality'? If you concede, perhaps ever so reluctantly, that the theories of physics have at least some relationship to 'reality', then I guess you've no choice but to conclude that 'infinities' do, indeed, occur in nature (and, as the above illustrates, in your everyday life).

Another example - much more open and shut (perhaps).

'Heat capacity (http://en.wikipedia.org/wiki/Heat_capacity)' is a pretty darn concrete thing, right? I mean, for goodness sake, you can measure it! And, since you can't measure 'infinity', there's no way that could possibly be infinitite, in 'reality', right? Wrong (http://en.wikipedia.org/wiki/Phase_(matter)).

Let's start with temperature.

We all know that F(ahrenheit) and C(elsius) are quite arbitrary - both for the zero point and the scale. Along came K(elvin), which at least has a non-arbitrary zero point, even if its scale is still arbitrary.

We've no doubt read about how close to 0K researchers have been able to get, and maybe we've read about a way to 'show' how unattainable 0K is, by taking the log of the temperature (0K then becomes -∞, which is, 'of course', unattainable).

But why do we use a linear scale for temperature? Why don't we use a logarithmic one? After all, such a scale would 'show' the unattainability of '0K' much more clearly, wouldn't it? And anyway, we already have several logarithmic thingies, don't we? pH, and decibels, and earthquakes (think Richter), for example.

It turns out that there is an interesting aspect to temperature and logs ... and it has to do with entropy. In one definition of temperature, it is just dE/dS, where E is the total energy of the system, and S the entropy; and S is just the logarithm of the number of microstates of the system.

And this definition leads to the possibility of negative temperatures (http://en.wikipedia.org/wiki/Negative_temperature), which, despite what you might think, are not colder than 0K, but hotter than +∞!

So, where might you encounter an infinity (temperature in this case) in your daily life? The CD/DVD player in your PC likely has a component that goes through a state which has a temperature of +∞ (and one with a temperature of -∞) during its normal operation. So does the barcode scanner in your local supermarket. And a great deal of other circumstances too.

But this is just a mathematician's (or physicist's) trick, right? Nothing is really infinite in nature, right?

Well, I guess that's up to you to decide - how does 'infinity', a mathematical concept, relate to 'reality'? If you concede, perhaps ever so reluctantly, that the theories of physics have at least some relationship to 'reality', then I guess you've no choice but to conclude that 'infinities' do, indeed, occur in nature (and, as the above illustrates, in your everyday life).

Another example - much more open and shut (perhaps).

'Heat capacity (http://en.wikipedia.org/wiki/Heat_capacity)' is a pretty darn concrete thing, right? I mean, for goodness sake, you can measure it! And, since you can't measure 'infinity', there's no way that could possibly be infinitite, in 'reality', right? Wrong (http://en.wikipedia.org/wiki/Phase_(matter)).