The Definition of Speed pt.I

I have written about my misgivings about the original, now somewhat discarded, 150 year old definition of entropy - and its consequences.

Those consequences being, in part:

The inability to come to grips with reality without throwing exception after exception, having to define such a thing as "negative" entropy to describe something positive, and getting into contradiction after contradiction and argument after argument when trying to define physical processes and conditions.

What, for instance, correctly and unequivocally defines thermodynamic closure? What are the universal conditions for the Second Law of Thermodynamics? How is entropy defined at low temperatures? At the beginning of the universe?

And so on.

The original definition

My take is that things would have been and would be different, in the general concept of the world, if Rudolf Clausius had defined entropy the other way around;

not

S=Q/T      (Quantity of heat by / Temperature),

but

S=T/Q      (Temperature by / Quantity of heat),

As division by zero is forbidden, the factor that may theoretically and independently reach that value (in this case, T) generally should not be put in the denominator of a fraction.

However, this was done; and more than just once.

Why?

The general situation

Starting from a blank state, defining a relationship between two factors as one divided by the other can be done in two, inverse ways; and this without misinterpreting reality, while laden with dire consequences in both cases.

So the question is primarily a question of practicability - with a profound effect on the perception of nature.

You could, for instance, describe the number of chairs (4) to a table (1) as 1 table to 4 chairs or 4 chairs to 1 table.

Or, a bit more intricate, the volume of air in a room (m³) per person (p) as  

 

a) m³/p or  

 

b) p/m³ 

 

(using m³ to visualize)

• The first equation (a) is not defined for 0 persons (and rightly so), while the second (b) is ("there is no-one in the room").
  
  
• In case of the volume of air being 0, the first expression spells suffocation or no room, while the second makes no sense.

Of course, in a room of fixed dimensions, people entering or leaving are the more independent variable, and can easily reach zero.

So it seems clever not to take them as the denominator, and define the relationship as persons per volume or p/m³ instead.

However - and that is the point - this is not done.

The necessary or resulting amount of air per person is usually expressed in as volume per person or m³/p - and never mind the zero, as any statement concerning an empty room is useless, and sensible use of the equation begins with the first person ("1") entering it.

Of course, one could enhance the expression by calculation 0.5 persons (children) or pets (0.25 persons) all the way down to one amoeba, but that only proves the point:

Although mathematically silly, we use the "wrong" one of the two possible ways to express a relationship between two variables.

Why?

Because it is easier, more intuitive and practicable in everyday life, and never mind the fringe silliness.

But more importantly, the question is:

Can you define the zero condition? 

A more technical example

Wikipedia states that the current definition of speed was first arrived at by Galileo Galilei, who laid down that "speed" (v) should furthermore be defined as "distance by time", or:

v = d / t

Now, it is of minor importance if it was indeed Galileo, or someone else, or when speed was defined exactly. The importance lies in the fact that:

1. This was a decision to make; the relationship between space and time could have been set in two ways:
   
   v = d / t ( = distance by time )   
   or   
   v = t / d ( = time by distance )
   
   
2. Before one definition was chosen over the other for all future, physical speed, other than "entropy" some centuries later, was not a new concept; it had been an issue over millennia, in the realms of sports, military, commerce and many more.
   
   
3. That said, since it obviously up to a certain point in history had not been defined in a mathematical equation, how had it been defined before?
   
   And had this historical definition been enhanced or discarded by Galileo, if we credit him with the new mathematical definition?
   
   
4. And why did he favor "distance by time" over "time by distance"?

We will have to speculate

There are two possibilities:

1. Galileo formulated the definition as it was being used, by giving contemporary common sense a mathematical expression.
2. He revolutionized science (as he was wont to) by defining it contrary to contemporary common sense.

This is just for clarification; the historical common sense definition of speed is of less importance than the consequences each definition would have.

And there are, again, two options for that definition:

v = d / t   or   v = t / d

Let's have a look at these two options