Who was the first person to decide properties like mass, energy, speed, and force can be multiplied and divided and even squared? What does it actually mean to use a property like it was a number? Isn't a quality something fundamentally different from a quantity? Or are we talking magnitudes used simply as variables? Please Register or Log in to view the hidden image!
I agree with Origin that the answer seems obvious...at first. But a property must be measurable and quantifiable before we can apply math to it, yes? We can measure colors, for example, but until we understood wavelengths of light we had no objective way to apply math to those values. What the hell is red squared? And is blue > yellow?
Precisely. And what does it even mean to multiply say mass by acceleration to get force? By "multiply" I assume we mean the arithmetical sense of the number of times a value is added to itself. So why is force equal to mass added to itself "acceleration" number of times? Or why does energy equal mass added to itself the speed of light times squared? What is speed of light squared anyway? I can see how a certain QUANTITY of a property can be multiplied or squared. But the property or constant itself? What does that even mean?
This...I believe I can answer. Physics is all about dimensions and how they relate to one another. I have this "thing" called position and this "thing" called time; well, let's say I can associate the position of a particular object with time; I can relate the dimensions to each other mathematically. With that I can calculate things like velocity, acceleration, etc. Basically all of our equations are really just taking a handful of these fundamental dimensions and making various sandwiches out of them (e.g. Force, etc) to give us higher levels of understanding about their complexities. Of course it all relies on the assumption that there is a mathematical relationship between, for example, an object's position and time in the first place. If objects had non-discrete and/or non-continuous positions with regard to time (such as quantum systems) then applying particular physics treatments to them would be futile...
One math operation that is useful but raises a yellow flag is division by a fraction. If I have 1 kg of mass, and divide this by 1/2, I now have 2 kg of mass. What exactly did I do here, in that I was able to create mass. The process of dividing by 1/2 did not even require the energy =mc2, where m-1 kg. I was able to do it with much less energy. In the miracle of the multitudes, Jesus feeds thousands with a few fish and few loaves of bread. in terms of math, he used division by a fraction. ALl he had to do was divide by 1/1000.
You can't divide 2 kg with a number [1/2] and wind up with 4 kg. Really dude? How many apples do you get if you divide 2 apples by 1/2? How many 1/2 apples do you get when divide 2 apples by 1/2 apple? Ever hear of dimensional analysis?
One way to see a disconnection between physics and the mathematics used to describe it, is that derivatives like dx/dt are axiomatic; mathematically dx/dt is not zero if x is some function of t. It seems obvious that a change in position 'includes' an interval of time.
Yes, time has been given a special treatment in Physics and unraveling that point was difficult for me, as a non-Physicist, because it was never explained. We could just as easily study, say, temperature with respect to position* of an object and then analyze the derivatives of this temperature/position relationship...it was simply a matter of convention and convenience that we try to associate properties to change in time. * Newton's law of cooling, for example, could be thought of as relating the temperature of an object with the position of the hands of a clocking device.
I think your basic question is about "multiplying units" and whether it makes sense or not. If the unit is "length", multiplication will make geometrical sense. For other units, it may make some equivalent sense which can be abstract also.
Excellent insight. That makes it much clearer. But re: these dimensions..To what extent are they just artificial constructs made of certain ways of measuring a certain magnitude? The mathematical relationships between the dimensions imparts to them a certain objective reality as if they are just out there on their own waiting to be discovered. But aren't the dimensions still just conceptual abstractions dependent on certain arbitrary standards and criteria we are applying to our experience. Like your color example. We have a pretty much continuous "analog" property there. We could quantify it into units of color measurement. Color suddenly equals these units times another variable say like wavelength. But how much of this mathematical structure comes from our own quantitative model of color? Aren't we really just finding equalities between dimensions we have already just mathematically projected upon the world?
I'd say it all comes down to practicality. Declaring that "the position of this train has a very close approximation to the mathematical formula f(x) = 3t" then it lets us make predictions based on that declaration. If our declaration is not accurate then our predictions will break down. Squaring colors doesn't really provide any practical use that I can immediately think of, but squaring length might be useful when, say, ordering fabric or some such...
It is really that nothing with units like mass, length etc has fundamental physical meaning, since our choice of units is artibrary. Why pick metres over yards over furlongs? Something which has no units is better, hence why things like the fine structure constant are unitless. Likewise you cannot take the sin, cos, tan, exp or log of a quantity with units, they must be dimensionless. This can be seen from their Taylor expansion, \(\exp(x) = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}\). If x has units of (for instance) length, then \(x^{2}\) has units of length squared, etc. and adding them is meaningless. Given that any function can be expressed as a Taylor series (pipe down those of you thinking "non-analytic functions!") of the form \(f(x) = \sum_{n=0}^{\infty}f^{(n)}(y)\frac{(x-y)^{n}}{n!}\) then it follows that ANY function of a quantity must only be applied to dimensionless quantities. The expressions used in physical formulae must satisfy this underlying principle if they are rearranged in the right way. In reality equations are often written in the form of most convenience (particularly in engineering domains) so something like F=ma having units on both sides is not troubling to anyone. In terms of historical development it comes from experiments. If the force a spring applies is twice as much when it is compressed by twice the amount that tells you something about how \(F \propto x\). If a ball dropped from 4 times as high only takes twice as long to hit the ground then you have \(d \propto t^{2}\). Once you have got all of these proportionality relationships then you pick your preferred units and put in the appropriate proportionality constants. For example, if you have worked out that the gravitational force between two objects of masses \(M_{1},M_{2}\) a distance r apart goes like \(F \propto M_{1}\), \(F \propto M_{2}\), \(F \propto \frac{1}{r^{2}}\) then you have \(F = G\frac{M_{1}M_{2}}{r^{2}}\) and you determine G in your preferred units. Of interesting note is that you can deduce \(F = G\frac{M_{1}M_{2}}{r^{2}}\) using a few reasonable assumptions without ever having to do an experiment. It follows from dimensional analysis.
This bewildered me a bit. I suppose, since the dimensions in an equation must cancel each other out by definition of equation, that any mathematical analysis could be done without dimensions...but does this mean we can't use dimensions in mathematical functions? Could you elaborate on this? I'd love to see the minimal list of assumptions to deduce that equality.
Yes, you must always be able to rearrange an equation to make it dimensionless, though few people ever bother for practical purposes. If I have something which is twice as massive as another object I shouldn't care if its 2kg vs 1kg or 100 billion nano-grams vs 50 billion nano-grams or whatever, all that matters is their ratio, which is dimensionless, is 2 (or 1/2, depending on your perspective). The book which does it properly (rather than me giving a less than clear 'off the top of my head' version) I've left in my office but I'll write it out sometime this week. In the mean time here's plenty of sites and pdfs which go through the application of the Buckingham Pi theorem (which is the underlying method we're talking about) for things like drag forces or pendulum oscillations.
It means that if you and a friend stand close together on a slippery surface, then push each other apart, then the lighter of you will accelerate away faster. The ratio of your accelerations will be the same as the ratio of your masses. Please Register or Log in to view the hidden image!
